# Year 09 - Mathematics - Unit 9

Mathematics (9-1) - iGCSE 2018-20 Year 09 By finding a counter-example, you can disprove statements such as 'All families in the UK spend less than an hour a day together. Which person provides a counter-example to each statement? All people wear a hat.' 'None of the people wear glasses.' Give a counter-example to show that this statement is false. All square numbers are even.' Unit 17 More Algebra Contents 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17 17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s More algebra

Prior knowledge check Rearranging formulae Algebraic fractions Simplifying algebraic fractions More algebraic fractions Surds Solving algebraic fraction equations Functions Proof Problem-solving: Surface gravity Check up Strengthen Extend Knowledge check Unit test Page Page iv iv 531 531 532 533 535 537 539 541 543 545 548 549 550 553 555 556 Contents 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving

Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page vi vi 17 Prior Knowledge Check 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction

Equations Surds Proof Surds Fraction Equations s s Numerical fluency 1. Find the LCM of 21 and 28. 2. Work out a. b. + c. x d. 3. Simplify these surds a. b. Algebraic fluency 4. Expand and simplify. a. 7(2 - 5x) b. (x + 4)(2x 3) Page Page 531 531 17 Prior Knowledge Check 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Algebraic fluency

5. Solve these equations. a. = 12 b. 7p - 3 = 3p + 17 c. 4(d + 5) = 7(d - 1) 6. Make x the subject of each formula. a. y = 4x + 7 b. W = h + 3hx c. y = 4(x + 1) d. P = Page Page 531 531 17 Prior Knowledge Check 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Algebraic fluency 7. Simplify a. y3 x y5 b. 4y2 x 7y4 c. y8 y d. 10y7 25y2 8. Factorise a. x2 + 6x +5 b. x2 7x - 30 c. x2 5x + 6

d. x2 - 36 Page Page 531 531 17 Prior Knowledge Check 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 9. Solve these equations by factorising, a. x2 + 11x + 30 = 0 b. x2 -12x + 11=0 c. 2x2 + 9x + 7 = 0 10. Solve x2 + 5x + 2 = 0 by using the quadratic formula. Leave your answer in surd form. 11. Solve x2 + 8x +10 = 0 by completing the square. Leave your answer in surd form. Page Page 531 531 17 Prior Knowledge Check 17.1

17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s * Challenge 12. a. Write down any five consecutive integers. b. Work out their sum. c. Repeat parts a and b for four more sets of consecutive integers. d. What do you notice? e. Predict the missing number in this sentence. The sum of five consecutive numbers is. multiple of ____. Use algebra to show why this happens. Page Page 531 531 Q12e hint Let the numbers be n, n+1, n + 2 17.1 Rearranging Formulae 17.1 17.1 Rearranging

Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 532 532 Objectives Why Learn This? Change the subject of a Physicists formula where the power of the rearrange subject appears. complex formulae Change the subject of a in order to find formula where the subject important appears twice. measures. Fluency y = 5x - 2. Find the value of x when y=3 y = -7 y=0 ActiveLearn - Homework, practice and support: Higher 17.1 17.1 Rearranging Formulae 17.1 17.1 Rearranging Rearranging 17.2 17.2

Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 532 532 Warm Up 1. In each formula change the subject to the letter given in brackets, a. v = u + at (a) b. C = 2 r (r) c. A = bh (h) d. A = r2 (r) e. x = (t) f. r = (s) 2. Factorise a. xy + 4y b. pq - q c. ak - 4k Questions in this unit are targeted at the steps indicated. 17.1 Rearranging Formulae 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions

Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 3. Make v the subject of the formula E = mv2 Q3 hint - First multiply both sides by 2. 4. Make x the subject of the formula H = Q4 hint - First square both sides. Page Page 532 532 17.1 Rearranging Formulae 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4

17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 532 532 Example Example 11 Make x the subject of the formula P = d = = = x or x = Divide both sides by d. Square both sides. 17.1 Rearranging Formulae 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6

17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 532 532 5. Make x the subject of each formula. a. T = 2p b. 4 c. P = d. L = 3(1 + x)2 Q5d hint - First divide both sides by 3. Then square root both sides 6. In each formula change the subject to the letter given in brackets. a. V = pr3 (r) b. V = 4x3 (x) c. y = (x) d. z = (y) Q4 Q4 hint hint -- First First make make rr33 the the subject. subject. Finally Finally take take the the cube cube root root to to give give rr as as the the subject. subject. Q5c Q5c hint hint -- First First cube cube both both sides. sides. 17.1 Rearranging Formulae 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic

Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 533 533 Key Point 1 When the letter to be made the subject appears twice in the formula you will need to factorise. Example 2 Make w the subject of the formula A = wh + lh + lw A lh = wh+ Iw A-lh = w(h + l) w = w appears twice in this formula. Subtract Ih from both sides to get the terms in w together on one side of the equals sign. Factorise the right-hand side, so w appears only once. Divide both sides by (h + l) 17.1 Rearranging Formulae 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae

Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 7. Make y the subject of the formula h = 3y + xy 8. Make d the subject of the formula H = ad ac - bd 9. Reasoning 5xy + 2 = w + 3xy a. Make y the subject, b. Make x the subject. Discussion What do you notice about your answers? 11. Make x the subject of the formula V= . Page Page 533 533 Q11 hint - First multiply both sides by x. 17.1 Rearranging Formulae 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3

Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 533 533 10. Reasoning H = xy + 2x + 7 Zoe rearranges the formula to make x the subject. Her answer is x = a. Explain why this cannot be the correct answer, b. What mistake has Zoe made? c. Work out the correct answer. 12 Exam-Style Questions Make k the subject of the formula t= (4 marks) Exam Exam hint hint First First multiply multiply both both sides sides by by (k (k -- 2). 2). Then Then expand expand the the bracket bracket on on the the leftlefthand hand side. side. 17.2 Algebraic Fractions

17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Objectives Add Add and and subtract subtract algebraic algebraic fractions. fractions. Multiply Multiply and and divide divide algebraic algebraic fractions. fractions. Change Change the the subject subject of of a a formula formula involving involving fractions fractions where where all all the the

variables variables are are in in the the denominators. denominators. Fluency Fluency Simplify Page Page 533 533 Why Learn This? Bridge Bridge designers designers use use algebraic algebraic fractions fractions when when making making sure sure their their designs are designs are structurally structurally safe. safe. ActiveLearn - Homework, practice and support: Higher 17.2 17.2 Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6

17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 1. Work out: a. + b. + 2. Work out: a. x c. b. Warm Up c. - Page Page 534 534 17.2 Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof

Fraction Equations Surds Proof Surds Fraction Equations s s 3. Write as a single fraction in its simplest form. The first one has been started for you. a. x = = b. x c. x Q3b Q3b hint hint -- First First cancel cancel any any common common factors factors 4. Write as a single fraction in its simplest form. The first one has been started for you. a. x = = b. x c. x Page Page 534 534 17.2 Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving

17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 5. Write as a single fraction in its simplest form. a. b. x3y c. d. Q5a hint Dividing by us equivalent to multiplying by Q5b hint Write x33y as Page Page 534 534 17.2 Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page

Page 532 532 Example 3 Simplify x LCM of 5 and 3 is 15 x3 = x3 + = x3 = x3 Find the LCM denominators. Write both fractions with the same denominator. Add the fractions 17.2 Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 6. Write as a single fraction in its simplest form: a. + b. - c. 7. Write down the LCM of a. 2x and 5x b. 3x and 6x c. 4x and 7x d. 4x and 3x

Q7a hint Multiples of 2x: 2x, 4x, ... Multiples of 5x: 5x, 10x, .. 8. a. Write and as equivalent fractions with denominator the LCM of 4x and 3x. b. Simplify and Page Page 534 534 17.2 Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 9. Write as a single fraction in its simplest form: a. + b. - c. + 10. a. Copy and complete: = = b. = = c. Use your answers to parts a and b to work out + Page Page 534

534 17.2 Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 535 535 11. Write as a single fraction in its simplest form. a. + b. c. 12 Exam-Style Questions Write as a single fraction in its simplest form. + (3 marks) Q12 Q12 strategy strategy hint hint -- Start Start by by rewriting rewriting each each fraction fraction

so so that that the the denominator denominator of of each each is is the the same. same. 17.2 Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 13. Make a the subject of the formula + = 1. The working has been started for you. + =1 =1= - = Q13 hint Write the right-hand side as a single fraction using the common denominator of 1 and b: b. Then find the reciprocal to find a. Page Page 535 535

17.2 Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 14. STEM Scientists use the lens formula to solve problems involving light. The lens formula is = + , where f = focal length, u = object distance and v = image distance. Make u the subject of the formula. Page Page 535 535 17.3 - Simplifying Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions

Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 533 533 Objectives Why Learn This? Simplifying Algebraic Fractions Aerospace engineers use and simplify algebraic fractions when designing planes. Fluency Factorise ActiveLearn - Homework, practice and support: Higher 17.3 17.3 - Simplifying Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae

Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 1. Simplify a. b. Warm Up 2. Fully factorise a. x2 9x + 18 b. x2 81 c. 5x2 + 21x + 4 Page Page 535 535 17.3 - Simplifying Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 3. Simplify a. c. 17.3 17.3

Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 535 535 b. d. Q3 hint - You You can can simplify simplify an an algebraic algebraic fraction ii the the same same way way as as simplifying simplifying aa normal fraction. Cancel any common factors in the numerator and denominator. Q3c hint - You can only cancel whole brackets. 17.3 - Simplifying Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying

Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Key Point 2 You You may may need need to to factorise factorise before before simplifying simplifying an an algebraic algebraic fraction: fraction: 1) 1) Factorise Factorise the the numerator numerator and and denominator. denominator. 2) 2) Divide Divide the the numerator numerator and and denominator denominator by by any any common common factors. factors. 4. a. Factorise x2 - 6x b. Use your answer to part a to simplify: Q4b hint - Replace Replace the the numerator numerator with

with your your factorisation factorisation from from part part a. a. Cancel Cancel common common factors. factors. Page Page 535 535 17.3 - Simplifying Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 5. Simplify fully: a. 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s b. Q5c hint - Factorise the numerator and denominator. 6. Reasoning Simplify Sally says, '(x + 2) is a factor of the numerator and the

denominator.' a. Is Sally correct? Explain. b. Can the fraction be simplified? Explain your answer. Page Page 536 536 17.3 - Simplifying Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 5. Simplify fully: a. 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s b. Q5c hint - Factorise the numerator and denominator. 6. Reasoning Simplify Sally says, '(x + 2) is a factor of the numerator and the denominator.' a. Is Sally correct? Explain. b. Can the fraction be simplified? Explain your answer. Page Page 536

536 17.3 - Simplifying Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 536 536 Example 4 Simplify fully = = 9 Exam-Style Questions Simplify fully Factorise the numerator and denominator Divide the numerator and denominator x 1 by the common factor (x + 4). Exam Exam hint hint First

First factorise factorise the the numerator numerator and and denominator. denominator. Use Use the the fact fact that that aa22 -- bb22 == (a (a ++ b)(a b)(a -- b). b). 17.3 - Simplifying Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 7. Simplify fully: a. 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s b. Q7a hint - Do not expand the numerator.

8. Simplify fully: a. c. Q8c hint Factorise (x22 25) using the difference of two squares. Page Page 536 536 17.3 - Simplifying Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 10.Simplify fully: a. 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s b. c. 11 Exam-Style Questions Simplify fully: (3 marks) Exam Exam hint hint -- 11 mark mark is

is awarded awarded For For correctly correctly factorising the 11 mark June 2012, Q23a, 1MA0/1H factorising the numerator; numerator; mark for for factorising factorising the the denominator; denominator; and and 11 mark mark for for the the correct correct Final Final answer. answer. Page Page 536 536 17.3 - Simplifying Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds

Proof Surds Fraction Equations s s 12. a. Copy and complete: (6 - x) =-( - ) b. Simplify i. ii. 13. Simplify fully: a. Page Page 536 536 17.3 - Simplifying Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 537 537 14. Communication Show that =Q14 hint - Start with the numerator and then the denominator.

Use factorising and simplifying to work towards -. 17.4 - More Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 537 537 Objectives Why Learn This? Add Add and and subtract subtract more more complex complex algebraic algebraic fractions. fractions. Multiply Multiply and and divide divide more more complex

complex algebraic algebraic fractions. fractions. Opticians Opticians use use algebraic algebraic fractions fractions when when working working out out a a lens lens prescription. prescription. Fluency Fluency Simplify: ActiveLearn - Homework, practice and support: Higher 17.4 17.4 More Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds

Proof Surds Fraction Equations s s Warm Up 1. Write as a single fraction a. b. c. 2. Write as a single fraction in its simplest form. a. + b. c. + Page Page 537 537 17.4 More Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 537 537

3. Write as a single fraction in its simplest form. a. (x + 3)2 x b. x c. x d. e. f. Key Point 3 You You may may need need to to factorise factorise the the numerator numerator and/or and/or denominator denominator before before you you multiply multiply or or divide divide algebraic algebraic fractions. fractions. Q3c hint - First factorise 3x - 12. 17.4 More Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6

17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 4. a. Factorise x2 9 b. Factorise x2 + 5x + 5 c. Write x as a single fraction in its simplest form. 5. Write as a single fraction in its simplest form. a. x b. Page Page 537 537 17.4 More Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations

s s 6. Write down the LCM of a. x and x + 2 b. x + 2 and x + 3 c. x + 4 and x + 5 d. x + 1 and x - 1 e. 2x - 3 and 2x - 4 Page Page 537 537 17.4 More Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Example 5 Write - as a single fraction in its simplest form. Common denominator = (x + 2)(x + 3) = Page Page 538 538

Factorise the numerator and denominator Convert each fraction to an equivalent fraction with the common denominator (x+2)(x + 3). Subtract the fractions = = Expand the brackets in the numerator, then simplify. 17.4 More Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 7. Simplify fully: a. + b. + c. d. 8 Exam-Style Questions Write as a single fraction in its simplest form: (3 marks) Exam Exam hint

hint Nov 2011, Q23c, 1380/4H Take Take care care when when multiplying multiplying out out aa bracket bracket which which has has aa negative negative sign sign in in front front of of it. it. Page Page 538 538 17.4 More Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 9. a. Factorise: i. 3x + 9

ii. 4x + 12 b. Write down the LCM of 3x + 9 and 4x + 12. c. Write + as a single fraction in its simplest form. Q9b hint - Look at the factorised form of each expression: a(x + y) b(x + y) LCM = ab(x + y) Page Page 538 538 17.4 More Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 538 538 10. a. Factorise x2 - 16 b. Write + as a single fraction in its simplest form.

11. Write as a single fraction in its simplest form. a. b. c. d. + Q11b Q11b hint hint -- Factorise Factorise (x (x22 ++ 7x 7x ++ 6) 6) and and (2x (2x ++ 12). 12). 17.4 More Algebraic Fractions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 12. Write + + as a single fraction in its simplest form. Q12 hint - Work out the lowest common denominator of 5x, 5(x - 1) and

10. 13. Communication Show that: + = and find the value of A. Page Page 538 538 17.5 - SURDS 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 539 539 Objectives Did You Know? Simplify Simplify expressions expressions Surds occur in involving involving surds. surds. nature. An example is Expand

Expand expressions expressions the Golden Ratio , involving involving surds. surds. which is also used in Rationalise Rationalise the the architecture. denominator denominator of of a a fraction. fraction. Fluency Fluency Are these numbers rational or irrational? -7 0.2 + ActiveLearn - Homework, practice and support: Higher 17.5 17.5 Surds 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions Q2a hint - Use x = Q2c hint - Use = Warm Up 1. Workout a. x b. 7 - 4 c. 3 + 5 2. Copy and complete. a. = x b. = x c. = 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving

17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 539 539 17.5 Surds 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Warm Up 3. Find the value of the integer k. a. = x = k b. = k c. = 4. Rationalise the denominators. Simplify your answers if possible. a.

b. Page Page 539 539 17.5 Surds 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 5. Simplify a. i. ii. b. Use your answers to part a to simplify 3 -+ 7 6. Simplify a. 2 - 3 b. + 3 c. 5 - +4 Q6a hint - First simplify each surd. = k, l Page Page 539 539 17.5 Surds 17.1 17.1

Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 7. Simplify a. + 2 = 2 + 2 = 2( + ) b. 9 + d. 8. Expand and simplify: a. (4 + b. ( + 1)(4 + ) c. (6 - )(4 + ) d. (2 - )2 f. (7 + 2 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s c. 18 - e. (4 - )2 Q8d Q8d hint hint -- (2 (2 -- ))22 == (2 (2 -- )(2 )(2 -- )) Your Your answer answer should should be be in in the the form form aa -- b.

b. Page Page 539 539 17.5 Surds 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 9 Exam-Style Questions Expand (5 - )2. Write your answer in the form a + b y where a, b and c are integers. (2 marks) 11. Rationalise the denominators. The first one has been started for you. a. x = = b. c. Page Page 539 539 17.5 Surds 17.1

17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 10.a. Work out the area of each shape. Write your answers in the form a + b. b. Reasoning Would the perimeter of each shape be rational or irrational? Explain. Page Page 540 540 17.5 Surds 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3

17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 13. Reasoning a. Expand and simplify (3 + )(3 - ) b. Is your answer rational or irrational? c. How can you tell if your answer will be rational or irrational? d. Which of these will have rational answers when expanded? i. (7 + )(2 - ) ii. (7+ )(7 + ) iii. (7 + )(7 - ) Check by expanding the brackets. e. Rationalise the denominator of Q13e hint - Multiply the numerator and denominator by (7 - ). Page Page 540 540 17.5 Surds 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions

17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 12 Exam-Style Questions Given that = a + b, where a and b are integers, find the value of a and the value of b. (3 marks) Exam hint Make you multiply both parts of the Junesure 2011, Q22b, 1380/3H expression in the numerator by . Key Point 4 To rationalize the fraction , multiply by To rationalize the fraction , multiply by Page Page 540 540 17.5 Surds 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions

17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 14. Reasoning a. Rationalise the denominators. Give your answers in the form a or a b where a, b and c are rational a. b. c. d. e. f. 15. a. Solve x2 - 6x + 1 = 0 by using the quadratic formula. b. Solve the equation x2 +10x + 13 = 0 by completing the square, c. Solve the equation x2 -16x +8 = 0. Write all your answers in surd form. Page Page 540 540 Q14a Q14a hint hint -- Multiply Multiply the the Q15a numerator and and denominator denominator Q15a hint hint -- Simplify Simplify your your surd surd numerator by answer. by (1 (1 -- ).). answer. 17.6 Solving Algebraic Fraction Equations 17.1 17.1 Rearranging

Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 541 541 Objectives Why Learn This? Solve equations that involve algebraic fractions. Pharmacists use algebraic fraction equations to calculate the correct dosage when issuing medication. Fluency Find the LCM of x and 4 4x and x x + 3 and x + 2 ActiveLearn - Homework, practice and support: Higher 17.6 17.6 Solving Algebraic Fraction Equations

17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 1. Simplify a. (x + 3)(x - 2) x c. + 3. Solve by factorizing: a. x2 + 6x + 8 = 0 b. 2x2- 13x +11 = 0 c. 5x2 25x + 20 = 0 4. Solve 3x2 + 8x - 17 = 0 by using the quadratic formula. Give your answers correct to 2 decimal places. Warm Up b. (x + 6)(x + 4) x 2. Write as a single fraction in its simplest form. a. b. - Page Page 541 541 17.6 Solving Algebraic Fraction Equations 17.1 17.1 Rearranging

Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 5. Solve these equations. Give your answer as a simplified fraction: a. + = 4 b. - = 7 c. 8 = Q5a Q5a hint hint -- First First simplify simplify the the LHS LHS of of the the equation. equation. 6. Solve these quadratic equations a. c. = = 4 b. = = d. = Q6a Q6a hint hint -- First First multiply multiply both both sides

sides by by the the LCM LCM (5x) (5x) and and simplify. simplify. Then Then multiply multiply out out the the bracket bracket and and solve solve by by factorising.. factorising.. Page Page 541 541 17.6 Solving Algebraic Fraction Equations 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 541 541

Example 5 Solve: + = 2 + =2 =2 = =2 11x + 2 = 2(2x - 1)(x + 2) 11x + 2 = 4x2 + 6x 4 4x2 5x 6 = 0 (4x + 3)(x - 2) = 0, so either 4x + 3 = 0 or x - 2 = 0 The solutions are x = - and x = 2. Rewrite Rewrite the the LHS LHS using using the the common common denominator denominator (2x (2x -- 1)(x 1)(x ++ 2) 2) Add Add the the fractions fractions Expand Expand the the brackets brackets in in the the numerator numerator and and simplify simplify Multiply Multiply both both sides sides by by (2x (2x -- 1)(x 1)(x ++ 2). 2). Multiply Multiply out out the the brackets brackets and and simplify simplify the the right-hand right-hand side. side. Rearrange Rearrange into into the the form form 22 ax ax ++ bx bx ++ cc == 00 Solve

Solve by by factorising factorising 17.6 Solving Algebraic Fraction Equations 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 542 542 7. Copy and complete Sioned's working to solve: + = 1 + = 1(x + 1)(2x - 3) 3) Multiply Multiply all all the the terms terms by by the the common 3(2x -3) +2(x + 1) = (x + 1)(2x - common denominator denominator (x (x ++ 1)

1) (2x (2x -- 3) 3) and and simplify simplify 6x 9 + 2x +2 = Simplify Simplify both both sides sides Reflect Sioned has used a different method and and expand expand the the to the example. Which method do you prefer? brackets. brackets. Why? Rearrange Rearrange into into the the form form ax ax22 ++ bx bx ++ cc == 00 17.6 Solving Algebraic Fraction Equations 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof

Surds Fraction Equations s s 8. Communication a. Show that the equation + = 1 can be rearranged to give x2 -10x + 9 = 0 b. Solve x2 -10x + 9 = 0 Discussion How can you check your solution is correct? 11 Exam-Style Questions Find the exact solutions of x + = 12 Exam Exam hint hint Find the Find the exact exact solutions solutions of of means means that that you you (3 marks) should should not not use use aa calculator. calculator. You You should should give give your your answers answers using using simplified simplified surds. surds. Page Page 542 542 17.6 Solving Algebraic Fraction Equations 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic

Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 9. Solve these quadratic equations. a. + = 1 b. + = 1 c. - = 1 d. - = 1 e. - = 1 Page Page 542 542 17.6 Solving Algebraic Fraction Equations 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction

Equations Surds Proof Surds Fraction Equations s s 10. Solve these quadratic equations. Give your answers correct to 2 decimal places. a. = b. + =5 c. - =1 d. + =3 Q10 communication hint - Give your answers correct to 2 decimal places' shows that you will need to use the quadratic formula. Page Page 542 542 17.7 Functions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations

Surds Proof Surds Fraction Equations s s Page Page 543 543 Objectives Why Learn This? Use function Function notation is an easy notation. way to distinguish different Find composite equations; each can be functions. labelled using different Find inverse letters. functions. Fluency x y X3 What squared is the output when x = 3 x = -2 x = 5? ActiveLearn - Homework, practice and support: Higher 17.7 17.7 Functions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction

Equations Surds Proof Surds Fraction Equations s s Warm Up 1. Write each expression using function machines. a. 2x + 5 b. - 6 c. 3(x + 1) 2. Find the value of x when a. 5x 3 = 4 b. 7x - 8 = 8 Page Page 543 543 17.7 Functions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 3. a. H = 4x and x = 3t. Write H in

terms of t. b. P = and x = y. Write P in terms of y. c. y = x2 and x = h + 3. Write y in terms of h. Q3a hint - Substitute x = 3t into H = 4x. Key Point 5 A A function function is is aa rule rule for for working working out out values values of of yy for for given given values values of of x. x. For For example, example, yy == 3x 3x and and yy == xx22 are are functions. functions. The The notation notation f(x) f(x) is is read read as as 'f'f of of x', x', ff is is the the function. function. f(x) f(x) == 3x 3x means means the the function function of of xx is is 3x. 3x. Page Page 543 543 17.7 Functions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying

Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 4. f(x) = . Work out: a. f(5) b. f(-2) c. f() d. f(-20) Q4a hint - Substitute x = 5 into . 5. Reasoning h(x) = 5x2. Alice says that h(2) = 100. a. Explain what Alice did wrong. b. Work out h(2). Page Page 543 543 17.7 Functions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic

Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 6. g(x) = 2x3. Work out a. g(3) b. g(-1) c. g() d. g(-5) Q6 hint - Use the priority of operations. 7. f(x) = x + x3, g(x) = 3x2. Work out a. f(1) + g(1) b. f(4) g(2) c. f(2) x g(4) d. e. 2g(10) f. 3f(-1) - g(3) Q7e hint - First work out g(10) and then multiply the answer by 2. Page Page 543 543 17.7 Functions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6

17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 8. g(x) = 5x - 3. Work out the value of a when a. g(a) = 12 b. g(a) = 0 c. g(a) = -7 Q8a hint - g(a) = 5a - 3 = 12 Solve for a. 9. f(x) = x2 - 8. Work out the values of a when a. f(a) = 17 b. f(a) = -4 c. f(a) = 0 d. f(a) = 12 Q9c hint - Write your answer as a surd in its simplest form. Page Page 543 543 17.7 Functions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving

17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 10. f(x) = x(x + 3), g(x) = (x - 1)(x + 5). Work out the values of a when a. f(a) = 0 b. g(a) = 0 = =-2 d. for g(a) Q10a -- f(a) Q10ac.hint hintf(a) f(a) = a(a a(a ++ 3) 3) == 0. 0. Solve Solve for a. a. = -8 11. f(x) = 5x - 4. Write out in full a. f(x) + 5 b. f(x) - 9 c. 2f(x) d. 7f(x) e.hint f(2x) Q11a - f(x)f.+f(4x) 5 = 5x - 4 + 5 = __ Q11c hint 2f(x) = 2(5x - 4) = _____ Q11e hint Replace x by 2x. Page Page 544 544 17.7 Functions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More

Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 12. h(x) = 3x2 - 4. Write out in full a. h(x) + 7 b. 2h(x) c. h(2x) d. h(-x) Discussion What do you notice about your answer to part d? Explain why this happens. Key Point 6 fg is a composite function. To work out fg(x), first work out g(x) and then substitute your answer into f(x). Page Page 544 544 17.7 Functions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8

17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 13. Reasoning f(x) = 6 2x, g(x) = x2 + 7. Work out a. gf(2) b. gf(7) c. fg(4) d. fg(5) Q13a hint - First work out f(2) and then substitute your answer into g(x). 14. Reasoning f(x) = 4x - 3, g(x) = 10 - x, h(x) = x2 + 7. Work out a. gf(x) b. fg(x) c. fh(x) d. hf(x) e. gh(x) f. hg(x) Q14a Q14a hint hint gf(x) gf(x) means means substitute substitute f(x) f(x) for for xx in in g(x). g(x). gf(x) gf(x) == g(4x g(4x -- 3) 3) == 10 10 -- (4x (4x -- 3) 3) == ___ ___ Page Page 544 544 17.7 Functions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions

Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 544 544 Key Point 6 The inverse function reverses the effect of the original function. Example 5 Communication hint hint -- xx Find the inverse function of x 5x - 1 Communication x X5 5 -1 5x - 1 + 1 x 5x 5x -- 11 is is another another way way of of showing showing f(x) f(x) == 5x 5x -- 11 Write Write the the function

function as as aa function function machine. machine. Reverse the function The inverse function of x 5x - 1 Reverse the function machine machine to to find find the the is x inverse inverse function. function. Start Start with with xx as as the the input input 17.7 Functions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 15. Find the inverse of each function. a. x 4x + 9

b. x - 4 c. x 2(x + 6) d. x 7(x 4) - 1 Q15a hint - You can check your answer by substituting e.g. x = 2 into the original function and then the answer into the inverse. Q15d hint Simplify the function first. x 7(x - 4) - 1 is the same as x 7x - 29 Page Page 544 544 17.7 Functions 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Key Point 8 f-1-1(x) is the inverse of f(x) 16. Reasoning f(x) = 4(x 1), g(x) = 4(x + 1) a. Find f-1(x). b. Find g-1(x). c. Work out f-1(x) + g1 (x). d. If f-1(a) + g-1(a) = 1 work out the value of a.

Page Page 544 544 17.8 Proof 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions Objectives 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 545 545 Did You Know? In the 1990s, Andrew Wiles spent over seven years trying Prove a to prove Fermat's Last result using Theorem. He received a algebra knighthood for his successful proof. Fluency What type of number is a. 2n b. 2n + 1 for any n?

ActiveLearn - Homework, practice and support: Higher 17.8 17.8 Proof 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Warm Up 1. Which sequences contain i. only even numbers ii. only odd numbers a. n + 2 b. 2n c. 5n d. 2n 1 e. n2 2. Expand and simplify, a. x(x -1) b. (x + 3)2 c. 2x(2x + 1) 3. Are these equations or identities? a. 2(n + 3) = 2n + 6 b. 5n - 7 = 8 c. (4n +10) = 2n + 5 Page Page 545 545

17.8 Proof 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 545 545 Key Point 6 To show a statement is an identity, expand and simplify the expressions on one or both sides of the equals sign, until the two expressions are the same. Example 8 Show that (x + 4)2 - 7 - x2 + 8x + 9 Expand the LHS = (x + 4) - 7 = (x + 4)(x + 4) - brackets on the 7 = x2 + 8x + 16 - 7 left-hand side = x2 + 8x + 9 (LHS). 2 RHS = x + 8x + 9 Aim to show that So LHS = RHS and (x + 4)2 - 7 LHS = RHS. x2 + 8x + 9 2

17.8 Proof 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 545 545 4. Communication Show that a. (x 3)2 + 6x x2 + 9 b. x2 + 8x + 49 (x + 7)2 - 6x c. (x - 5)2 - 4 (x - 3)(x - 7) d. 16 (x + 2)2 (6 + x)(2 - x) Reflect For part c can you think of a different method than the one given in the hint? Q4b hint Start with RHS Q4c hint First expand and simplify the LHS. Then factorise. 5. Communication / Reasoning a. Show that (x - 1)(x + 1) x2 - 1 b. Use your rule to work out i. 99 x 101 ii. 199 17.8 Proof

17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 545 545 6. Reasoning The blue card is a rectangle of length x + 5 and width x + 2. a. Write an expression for the area of the blue card. A rectangle of length x + 1 and width x is cut out and removed. b. Write an expression for the area of the rectangle cut out. c. Show that the area of the remaining card is 6x + 10. Q6c Q6c hint hint Subtract Subtract your your expression expression from from part part b b from from your

your expression expression from from part part a. a. 17.8 Proof 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 7 Exam-Style Questions The diagram shows a large rectangle of length (3x + 4)cm and width x cm. A smaller rectangle of length x cm and width 5 cm is cut out and removed. The area of the shape that is left is 70 cm2. Show that 3x2 - x - 70 = 0. (3 marks) Exam Exam hint hint Show Show that... that... means means you

you need need to to write write down down every every stage stage of of your your working. working. Page Page 546 546 17.8 Proof 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 8. Give a counter-example to prove that these statements are not true, a. All prime numbers are odd. b. The cube of a number is always greater than its square, c. The difference between two numbers is always less than

their sum. d. The difference between two square numbers is always odd. Q8a hint List some prime numbers Q8c hint Try some negative numbers. Page Page 546 546 17.8 Proof 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Key Point 10 A A proof proof is is aa logical logical argument argument for for aa mathematical mathematical statement. statement. To To prove prove aa statement statement is is true, true, you

you must must show show that that itit will will be be true true in in all all cases. cases. To To prove prove aa statement statement is is not not true true you you can can find find aa counter-example counter-example -- an an example example that that does does not not fit fit the the statement. statement. 8. Communication / Reasoning a. Prove that the sum of any odd number and any even number is always odd. b. Reasoning Explain why any odd number can be written as 2n +1 or 2n -1. Q9a hint Let 2n be any even number. Let 2n + 1 be any odd number. Page Page 546 546 17.8 Proof 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic

Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 10. Communication / Reasoning a. The nth even number is 2n. Explain why the next even number is 2n + 2. b. Prove that the product of two consecutive even numbers is a multiple of 4. 11. Communication / Reasoning Prove that the product of any two odd numbers is odd. 12. Communication / Reasoning Given that 2(x - a) = x + 5, where a is an integer, show that x must be an odd number. Page Page 546 546 17.8 Proof 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6

17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 13. Communication / Reasoning a. Work out i. ii. iii. b. Use your answers to part a to write down the answer to c. Reasoning Explain how you can quickly calculate d. i. Simplify ii. Reasoning Explain how this proves your answer from part c. Page Page 546 546 17.8 Proof 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s

14. Communication / Reasoning Show that: - = and find the value of A. 15. Communication / Reasoning Prove that n2 + n is a multiple of 2 for all values of n. Q15 hint Factorise first Page Page 547 547 17.8 Proof 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 547 547 16. Communication a. Write an expression for the product of three consecutive integers, n - 1, n and n + 1. b. Hence show that n3 - n is a multiple of 2. Q16b hint Consider when n is even

and when n is odd. 17 Exam-Style Questions Prove algebraically that the difference between the squares of any two consecutive integers is equal to the sum of these two integers. (4 marks) Q17 Q17 strategy strategy hint hint -- Start Start by by using using algebra algebra to to write write down down expressions expressions for for the the squares squares of of two two numbers numbers that that are are consecutive. consecutive. 17.8 Proof 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof

Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 548 548 Be able to rearrange equations that involve powers. Objectives Be able to use standard form in calculations. Big objects, like planets, create gravity that pull objects towards them. This is strongest on the surface of the planet. To find the surface gravity of a planet we can use the formula: The formula uses a gravitational constant. This is a number that was calculated many years after the formula was constructed. G = 6.67 x 10-11 g = where M = mass of the planet (kg), r = radius of planet (m), G = gravitational constant 17.8 Proof 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction

Equations Surds Proof Surds Fraction Equations s s Page Page 548 548 1. Earth's surface gravity is approximately 9.81 m/s2. The mass of Earth is 5.97 x 1024 kg. Find the radius of the Earth in kilometres (to 3 significant figures). Q1 Q1 hint hint -- Rearrange Rearrange the the equation equation to to make make rr the the subject subject 2. To find the volume of a planet we can assume that it is spherical and use the formula V= 3. The volume of Mars is 1.63 x 1020m3 a. Find the radius of Mars (to 3 significant figures). b. Given that the mass of Mars is 6.42 x 1023 kg, calculate the gravity on the surface of Mars. Q2a Q2a hint hint -- Rearrange Rearrange the the equation equation for for the the volume volume of of the the sphere sphere to to maker maker the the subject. subject. 17.8 Proof 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying

Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 548 548 3. The density of Saturn is 687 kg/m3. Its mass is 5.68 x 1026kg. Calculate the gravity at the surface. Q3 hint - Remember that Density = mass + volume 4. Imagine that the Earth starts growing. Assuming that the Earth's density remained constant, what radius would it need to grow to in order to have the same surface gravity as on Saturn? Q4 Q4 hint hint -- Start Start by by combining combining the the equation equation for for the the volume volume of of aa sphere, sphere, the the equation equation for for gravity gravity and and the the equation equation for for density. density. Remember Remember that that as as the the radius

radius increases increases so so will will the the mass. mass. Your Your equation equation will will only only involve involve values values that that will will remain remain constant constant (G, (G, D, D, gg and and r). r). 17 Check-Up 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 549 549

Log how you did on your Student 16. Check up Progression Chart Surds 1. Simplify a. + 2 b. (4-)2 2. Rationalise the denominators. a. b. Formulae and functions 3. Find f -1(x) for each function, a. f(2) = b. f(x) = 3x + 4 17 Check-Up 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 4. f(x) = 9 2x, g(x) = x2 + 4x. Work out a. f(2) + g(3) b. f(2) - g(3) c. f(3) x g(4) 5. Make y the subject of the formula z = 6. Make y the subject of 5xy + 3x = 9 2y

Page Page 549 549 17 Check-Up 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 7. Make k the subject of the formula T = 2p 8. f(x) = 4x2 - 7 a. Work out f(3). b. Find the value of a where f(a) = 0. Algebraic fractions 9. Simplify fully: a. b. Page Page 549 549 17 Check-Up 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic

Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 10. Write as a single fraction in its simplest form. a. b. + c. 11. Write as a single fraction in its simplest form. a. x b. Page Page 549 549 17 Check-Up 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions

17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 12. Solve the equation - =1 Proof 13. Communication Show that 23 - (x + 1)2 = (6 + x) (4 - x) 14. Communication / Reasoning Prove that this statement is not true: The sum of two cubed numbers is always odd. Page Page 549 549 17 Check-Up 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7

17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Reflect 15. How sure are you of your answers? Were you mostly Just guessing Feeling doubtful Confident What next? Use your results to decide whether to strengthen or extend your learning. * Challenge 16. Prove that a. the sum of two consecutive odd numbers is a multiple of 4 b. the sum of three consecutive even numbers is a multiple of 6 c. the sum of four consecutive odd numbers is a multiple of 8. Page Page 549 549 Q16 hint - Let 2n be any even number and 2n + 1 be any odd number. 17 Strengthen 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic

Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 550 550 17 - Strengthen 1. Copy and complete, a. x = b. x = 7 c. 2 x =0 d. (6 - ) = x 6 - x = 6 - e. + Q1d hint - x 6 = 6 x = 6 Always write the whole number before the surd Q1e hint - = and = 17 Strengthen 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving

17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 550 550 2. Rationalise the denominators. a. = x = = c. b. = x = = Q2a hint - x = 3. To get rid of in the denominator, multiply the fraction by Q2b hint -Multiply both parts of the expression in the numerator by Q2 Q2 communication communication hint hint -- 'Rationalise 'Rationalise the the denominator' denominator' means means get get rid rid of of any any surds surds in in the the denominator, denominator, so so itit is is aa rational rational number. number. 17 Strengthen 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions

17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 550 550 3. Expand and simplify. The first one has been started for you. a. (4 - )(2 + ) = 8 + 4 - b. (5 - )2 = + c. (3 - )(3 + ) = 9 + 3 - d. (2 + )(2 - ) e. (4- )(4 + ) f. Reasoning Look at your answers to parts c to e. What do you notice? Why does this happen? g. What would you multiply these expressions by to get an integer answer? Q3b hint (5 - )22 = i. (6 + ) ii. (3 - ) (5 - ). Q3 hint - Multiply each term in the second bracket by each term in the first bracket. FOIL: Firsts, Outers, Inners, Lasts. 17 Strengthen 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8

17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 4. Rationalise the denominators. a. = b. Q4a hint To get rid of (5 - ) in the denominator, multiply the fraction by: Page Page 550 550 17 Strengthen 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations

s s Formulae and functions 1. a. Copy and complete. i. y = , so y2 = ii. y = , so y2 = iii. y = , so y2 = b. Use your answer to part a iii to make x the subject of the formula y = Q1b hint Your answer will be x = _____ Page Page 550 550 17 Strengthen 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 551 551 Formulae and functions 2. Here are all the steps to make y the subject of

x = Match each step to one of these rearrangements. Rewrite the formula so there is no fraction. y(x - 1) = 7 Get all the terms containing y on the left-hand side and all other terms on the right-hand side. xy = 7 + y y= Factorise so that y appears only once. Get y on its own on the left-hand side. xy y = 7 17 Strengthen 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 3. Make y the subject of the formula F = Q3 hint - Follow the steps in Q2. 4. a. y = 5x - 9. Work out the value of y when x = 2. b. f(x) - 5x - 9. Work out f(2).

c. Work out: i. f(5) ii. f(-3) iii. f(0) Q4b hint - f(2) means substitute x = 2 in 5x - 9. Page Page 551 551 17 Strengthen 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 5. a. Solve these equations. i. 8x - 1 = 0 ii. 2 - 7x = 0 b. f(x) = 8x -1 and g(x) = 2 7x. Use your answers to part a to find the value of a where i. f(a) = 0 ii. g(a) = 0 c. p(x) = 9x - 4. Find the value of a where p(a) = 0. Q5b i hint - f(x) = 8x 1 f(a) = 0 means that 8a 1 = 0 Page Page 551

551 17 Strengthen 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 551 551 6. Reasoning f(x) = , g(x) = x2 - 1, h(x) = x(x - 5). Work out a. f(5) b. g(6) c. f(5) x g(6) e. 2g(6) f. g(2x) g. i. g(3) iii. f(2) iv. hf(2) d. 4f(5) ii. hg(3) Q6c hint Multiply tour value for f(5) Q6d hint 4f(5) by your value for g(6) means 4 x f()5 Q6f hint - g(x) = x22 -1 g(2x) = (2x)2 - 1 = __ - 1 Q6g ii hint Substitute

your value for g(3) into h(x) 17 Strengthen 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 7. Jake draws a function machine to illustrate y = 5x - 4. To find the inverse function he reverses the machine and replaces the functions with their inverse. x x5 -4 y y 5 +4

x Copy and complete the inverse function. y= Page Page 551 551 17 Strengthen 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 8. Find f-1(x) for each function. a. f(x) = 2x - 9 b. f(x) = 3(x - 5) c. f(x) = d. f(x) = Q8 hint - Use the same method as Q7 Page Page 551 551 17 Strengthen 17.1 17.1

Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Algebraic fractions 1. Simplify a. b. c. d. x e. x x f. x x Q1 hint Cancel common factors Q1a hint - Look to group common factors. - x Page Page 552 552 17 Strengthen 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic

Formulae Fractions Formulae Fractions 2. a. 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Copy and complete i. = ii. = iii. iv. b. Use your answers from part a to fully simplify x Q2b hint - Regroup the terms and cancel common factors. x = x x x Page Page 552 552 17 Strengthen 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions

17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 3. Write each of these as a single fraction in its simplest form. a. x b. Q3 strategy hint - Use the same strategy as in Q2. Q3b hint - Multiply the first fraction by the reciprocal of the second fraction: Page Page 552 552 17 Strengthen 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More

More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 4. a. Factorise i. 3x + 18 ii. x2 + 6x b. Use your answers from part a to simplify fully. c. Simplify fully: Q4b hint - Rewrite the numerator and denominator in factorised form. Cancel common factors. Q4c hint Use the difference of two squares. x22 25 = (x + 5)(x - 5) Page Page 552 552 17 Strengthen 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6

17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 5. Simplify fully: a. = = b. c. Page Page 552 552 17 Strengthen 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 6. a. Factorise:

i. 3x + 9 ii. x2 + 9x + 18 iii. x2 + 8x + 15 iv. 2x + 10 b. Use your answers to part a to write as a single fraction. i. x ii. Page Page 552 552 17 Strengthen 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 7. Solve these quadratic equations, a. (x - 8)(x + 7) = 0 b. x2 2x - 63 = 0 c. x2 + 3x + 3 = 4x + 9 d. = 1 Q7b hint - Factorise the equation Q7b hint - Rearrange the equation into the form x22 + bx + c = 0. Page Page 552

552 17 Strengthen 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 8. a. Write down the LCM of x and x -1. b. Copy and complete. i. = ii. = c. Copy and complete, using your answers to part b. = = = d. Use your answer to part c to solve = = 1 Q8b hint - First set your fraction answer from part c equal to 1. Page Page 553 553 17 Strengthen 17.1 17.1 Rearranging Rearranging 17.2

17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 9. Write as a single fraction in its simplest form. = Q9 hint - First factorise x22 - 5x + 4. Proof 10.a. Expand (x - 4)2 b. Expand and simplify (x - 4)2 - 9 c. Expand (x - 7)(x - 1) d. Use your answers to parts b and c to show that (x - 1)2 - 9 = (x - 7)(x - 1) Q1d hint means identical to Page Page 553 553 17 Strengthen 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions

17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 2. Communication Show that (x - 1)2 - 16 (x - 5)(x + 3) 3. a. List the first five cube numbers, b. Give a counter-example to prove this statement is not true: The difference between two cube numbers is always odd. Q3 hint Look for a pair of numbers in your list from part a whose difference is even. Page Page 553 553 17 Extend 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions

17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 17 - Extend 1. Communication / Reasoning Both Jack and Ruth make y the subject of the formula 1 - 2y = x Jacks answer is y = Ruth's answer is y = a. Show that both answers are correct. b. Explain why Ruth's answer might be considered a better answer. c. Make x the subject of the formula = d Page Page 553 553 17 Extend 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions

Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 2. STEM The total resistance of a set of resistors in a parallel circuit is given by the formula = + Make R2 the subject of the formula. 3. Communication = + a. Write down an expression for b. Show that d = Page Page 553 553 17 Extend 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof

Fraction Equations Surds Proof Surds Fraction Equations s s 4. Solve these equations. a. = b. - = 5 5. Communication a. Show that = b. Work out the exact value of Page Page 554 554 17 Extend 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 554 554 6 Exam-Style Questions

The functions f and g are such that f(x) = 3 4x, g(x) = 3 + 4x a. Find f(6) b. Find gf(x) c. Find i. f-1(x) ii. g-1 (x) d. Show that f-1(x) + g-1(x) = 0, for all values of x. Q6 Q6 strategy strategy hint hint ff-1-1(*) (*) is is the the inverse inverse of of f(x). f(x). 7 (7 Exam-Style marks) Questions F(x) = x + 7, g(x) = x2 + 6 a. Work out i. fg(x) ii. gf(x) b. Solve fg(x) - gf(x) Q7 strategy hint - Remember that fg means do g first and then f. (6 marks) 17 Extend 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations

Surds Proof Surds Fraction Equations s s 8. Communication / Reasoning f(x) = , g(x) a. Work out i. fg(x) ii. gf(x) b. Are f(x) and g(x) inverse functions? Explain your answer, c. Check whether f(x) = x - 1 and g(x) = 4(x + 1) are inverse functions. Q8 hint - Functions f and g are inverses of each other if fg(x) = gf(x) = x Page Page 554 554 17 Extend 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s

Page Page 554 554 9. Communication / Reasoning Show that = -1 10. Communication / Reasoning Show that (3n + 1)2 - (3n - 1)2 is a multiple of 12, for all positive values of n. 11. Communication / Reasoning Show - = and find the value of A and B. 17 Extend 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 554 554 12. STEM Newton's Law of Universal Gravitation can be used to calculate the force (F) between two different objects. F = , where G is the gravitational constant (6.67 x 10-11 Nm2kg-2), m1 and m2 are the masses of the two objects (kg) and r is the distance between them (km), a. Rearrange the formula to make r the subject. The gravitational force between the Earth and the Sun is 3.52 x 10 22N. The mass of the Sun is 1.99 x 1030 kg and the mass of the Earth is 5.97 x 1024kg. b. Work out the distance between the Sun and the Earth.

17 Knowledge Check 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 555 555 17 Knowledge Check You can change the subject of a formula by isolating the terms involving the new subject. When the letter to be made the subject appears twice in the formula you will need to factorise. To add or subtract algebraic fractions, write each fraction as an equivalent fraction with a common denominator. You may need to factorise before simplifying an algebraic fraction: o Factorise the numerator and denominator. o Divide the numerator and denominator by any common factors. To find the lowest common denominator of algebraic fractions, you may need to factorise the denominators

first. Mastery Lesson 17.1 Mastery Lesson 17.1 Mastery Lesson 17.2 Mastery Lesson 17.3 Mastery Lesson 17.4 17 Knowledge Check 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 555 555 17 Knowledge Check

You may need to factorise the numerator and/or denominator before you multiply or divide algebraic fractions. To rationalise the fraction multiply by A function is a rule for working out values of y when given values of x e.g. y = 3x and y = x2. The notation f(x) is read as 'f of x' fg is the composition of the function f with the function g. To work out fg(x), first work out g(x) and then substitute your answer into f(x). The inverse function reverses the effect of the original function. f-1(x) is the inverse of f(x). To show a statement is an identity, expand and simplify the expressions on one or both sides of the equals sign, until the two expressions are the same Mastery Lesson 17.4 Mastery Lesson 17.5 Mastery Lesson 17.7 Mastery Lesson 17.7 Mastery Lesson 17.7 Mastery Lesson 17.8 17 Knowledge Check 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof

Surds Fraction Equations s s Page Page 555 555 17 Knowledge Check Mastery A proof is a logical argument for a mathematical statement. To prove a statement is true, you must show that it will be Lesson 17.8 true in all cases. Mastery To prove a statement is not true you can find a counterLesson 17.8 example - an example that does not fit the statement. For an algebraic proof, use n to represent any integer. Even Number 2n Odd Number 2n + 1 or 2n 1 Consecutive numbers n, n + 1, n + 2, Consecutive even numbers 2n, 2n + 2, 2n + 4, Consecutive off numbers 2n + 1, 2n + 3, n + 5, Reflect Look back at this unit. Which lesson made you think the hardest? Write a sentence to explain why. Begin your sentence with: Lesson ____ made me think the hardest because _______ 17 Unit Test 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17 Unit Test 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4

More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 556 556 Log how you did on your Student Progression Chart. 1. Find the inverse of the function f(x) = 5(x + 4) (3 marks) 2. Show that (x + 4)2 - (2x + 7) (x + 3)2 for all values of x. 17 Unit Test 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6

Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 5. Write as a single fraction in its simplest form. a. x (4 marks) b. x 6. Make x the subject of the formula V = (2 marks) 7. Expand and simplify, Page Page 556 556 17 Unit Test 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof

Surds Fraction Equations s s 8. Write as a single fraction in its simplest form. (4 marks) a. b. x 9. Rationalise the denominator. (2 marks) 10. Solve these quadratic equations. (6 marks) a. = b. + = Page Page 556 556 17 Unit Test 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s 11. Solve the equation - = 1. Give your answers correct to 2 decimal places. (3 marks) 12. f(x) = x2 - 9, g(x) = 2x + 1 a. Work out i. f(4) + g(2)

ii. f(2) x g(-1) b. Find the value of a where f(a) = 0. c. Show that fg(x) = 4x2 + 4x 8 (5 marks) Page Page 556 556 17 Unit Test 17.1 17.1 Rearranging Rearranging 17.2 17.2 Algebraic Algebraic Formulae Fractions Formulae Fractions 17.3 17.3 Simplifying Simplifying Fractions Fractions 17.4 17.4 More More Algebraic Algebraic Fractions Fractions 17.7 17.7 -17.8 17.6 17.5 17.6 Solving Solving 17.5 -Function 17.8 Function Proof Fraction Equations Surds Proof Surds Fraction Equations s s Page Page 556 556 Sample student answer Explain what common mistake the student has made right at the very start of the answer. Suggest a way to avoid making this mistake. Exam-Style Questions Make b the subject of the formula a= (4 marks)

May 2008, Q22, 5540/3H J Student answer Ab - 5 = 2 - 7b ab + 7b = 2 + 5 b(a + 7) = 7 b=

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