First published 2014 by Heriot-Watt University. This edition published in 2017 by Heriot-Watt University SCHOLAR. Copyright 2017 SCHOLAR Forum.

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1 SCHOLAR Study Guide National 5 Mathematics Assessment Practice 3: Expanding brackets, factorising, completing the square, changing the subject of a formula and algebraic fractions (Topics 8-12) Authored by: Margaret Ferguson Heriot-Watt University Edinburgh EH14 4AS, United Kingdom.

2 First published 2014 by Heriot-Watt University. This edition published in 2017 by Heriot-Watt University SCHOLAR. Copyright 2017 SCHOLAR Forum. Members of the SCHOLAR Forum may reproduce this publication in whole or in part for educational purposes within their establishment providing that no profit accrues at any stage, Any other use of the materials is governed by the general copyright statement that follows. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, without written permission from the publisher. Heriot-Watt University accepts no responsibility or liability whatsoever with regard to the information contained in this study guide. Distributed by the SCHOLAR Forum. SCHOLAR Study Guide Assessment Practice: National 5 Mathematics 1. National 5 Mathematics Course Code: C747 75

3 Acknowledgements Thanks are due to the members of Heriot-Watt University's SCHOLAR team who planned and created these materials, and to the many colleagues who reviewed the content. We would like to acknowledge the assistance of the education authorities, colleges, teachers and students who contributed to the SCHOLAR programme and who evaluated these materials. Grateful acknowledgement is made for permission to use the following material in the SCHOLAR programme: The Scottish Qualifications Authority for permission to use Past Papers assessments. The Scottish Government for financial support. The content of this Study Guide is aligned to the Scottish Qualifications Authority (SQA) curriculum. All brand names, product names, logos and related devices are used for identification purposes only and are trademarks, registered trademarks or service marks of their respective holders.

5 1 Topic 7 Expanding brackets, factorising, completing the square, changing the subject of a formula and algebraic fractions Contents 7.1 Learning points Assessment practice

6 2 TOPIC 7. EXPANDING BRACKETS, FACTORISING, COMPLETING THE SQUARE, CHANGING THE SUBJECT OF A FORMULA AND ALGEBRAIC FRACTIONS By the end of this topic, you should have identified your strengths and areas for further revision. Read through the learning points before you attempt the assessments and go back to the Course Materials unit if you need more help. Key point You should be able to: multiply out single brackets; multiply out double brackets; factorise using a single common factor; factorise a difference of two squares; factorise trinomials; complete the square in quadratic expressions; change the subject of a formula; simplify algebraic fractions.

7 TOPIC 7. EXPANDING BRACKETS, FACTORISING, COMPLETING THE SQUARE, CHANGING THE SUBJECT OF A FORMULA AND ALGEBRAIC FRACTIONS Learning points Expanding brackets When expanding a single bracket remember that every item inside the bracket is multiplied by the term outside the bracket. When expanding double brackets remember that each term in the second bracket is multiplied by each term in the first bracket by turning the expression into two single bracket expressions or using the rainbow method or FOIL. Factorising When factorising always ask yourself three questions: 1. Is there a simple common factor? 2. Is it a difference of two squares? 3. Is it a trinomial? and remember you could have a simple common factor and a difference of two squares or a simple common factor and a trinomial. Completing the square Completing the square has a method which requires practice. You will need this to be able to find the turning point of a parabola. Always make the expression take the form ax 2 + bx + c. Write the expression in the form (x + k) 2 k 2 + c OR [(x + k) 2 k 2 ] + c. Tidy up your answer by calculating k 2 and expanding any square brackets. Changing the subject of a formula By reversing the operations performed in the original formula you can change the subject. The rules for solving equations can be applied. Re-arrange a step at a time and show all working.

8 4 TOPIC 7. EXPANDING BRACKETS, FACTORISING, COMPLETING THE SQUARE, CHANGING THE SUBJECT OF A FORMULA AND ALGEBRAIC FRACTIONS Algebraic fractions Simplifying algebraic fractions means reducing the numerical fraction to it's simplest form as well as the algebraic part e.g. 2 2 =1and x x =1, 3 6 = 1 2 and x = 1 x 2 x. Simplifying algebraic fractions may require factorising first. Before adding and subtracting algebraic fractions you must have a common denominator e.g. 2 x + 3 y = 2y+3x xy Multiplying fractions is the simplest, you multiply the numerators together then multiply the denominators together 2x 3y x 5 = 2x2 15y To divide by a fraction change the operation to multiplication and flip the second fraction. ie. Multiply by the reciprocal of the second fraction. 4x y x 3 = 4x y 3 x = 12x xy = 12 y Simplify the answer if necessary.

9 TOPIC 7. EXPANDING BRACKETS, FACTORISING, COMPLETING THE SQUARE, CHANGING THE SUBJECT OF A FORMULA AND ALGEBRAIC FRACTIONS Assessment practice Make sure that you have read through the learning points or completed some revision before attempting these questions. Tailor your practice by choosing the most appropriate questions. Expanding brackets: Questions 1 to 5 Factorising: Questions 6 to 11 Completing the square: Questions 12 to 14 Changing the subject of the formula: Questions 15 to 18 Algebraic fractions: Questions 19 to 23 Key point None of these questions assess your reasoning skills. Assessment practice: Expanding brackets, factorising, completing the square, changing the subject of a formula and algebraic fractions Go online Expanding brackets Expand and simplify these equations. Q1: 5c(c 3d) Q2: (y 3)(y +7) Q3: (2a + 1)(3a 2) Q4: 2(g 1)(g +5) Q5: (3x +1)(x 2 2x +5) Factorising Factorise these equations. Q6: 6k 2 9k Q7: j 2 81

10 Q8: 25m 2 36n 2 6 TOPIC 7. EXPANDING BRACKETS, FACTORISING, COMPLETING THE SQUARE, CHANGING THE SUBJECT OF A FORMULA AND ALGEBRAIC FRACTIONS Q9: x 2 +4x 12 Q10: 20x 2 80 Q11: 2x 2 x 10 Completing the square Q12: Express x 2 + 6x 2 in the form (x + p) 2 + q Q13: Express x 2 6x + 5 in the form (x p) 2 + q Q14: Express 1+6x x 2 in the form (x + p) 2 + q Changing the subject of the formula Q15: The equation of a straight line takes the form y = mx + c, make x the subject of the formula. 1 Q16: The formula for the volume of a cone is V = 3 πr2 h, make r the subject of the formula. Q17: The formula to find the equation of a straight line is y b = m(x a), make x the subject of the formula. Q18: Make x the subject of the formula ax by = cx Algebraic fractions Simplify these equations. Q19: (2x +3y) 2 (2x +3y)(x +2y) Q20: 5 h g 6

11 TOPIC 7. EXPANDING BRACKETS, FACTORISING, COMPLETING THE SQUARE, CHANGING THE SUBJECT OF A FORMULA AND ALGEBRAIC FRACTIONS 7 Q21: 2k k 2 2 Q22: b b 3 Q23: 4m2 n 2m n

12 8 ANSWERS: UNIT 2 TOPIC 3 Answers to questions and activities Topic 3: Expanding brackets, factorising, completing the square, changing the subject of a formula and algebraic fractions Assessment practice: Expanding brackets, factorising, completing the square, changing the subject of a formula and algebraic fractions (page 5) Q1: 5c 2 15cd Q2: y 2 +4y 21 Q3: 6a 2 a 2 Q4: 2g 2 +8g 10 Q5: Steps: What is 3x(x 2-2x + 5)? 3x 3-6x x What is 1(x 2-2x + 5)? x 2-2x+5 Collect like terms from these answers to simplify. Answer: 3x 3 5x 2 +13x +5 Q6: 3k(2k 3) Q7: (j 9)(j +9) Q8: (5m 6n)(5m +6n) Q9: (x 2)(x +6) Q10: Hints: Find the highest common factor first. Answer:20(x 2)(x +2) Q11: (2x 5)(x +2) Q12: x 2 +6x 2 = ( x 2 +6x ) 2 = (x +3) = (x +3) 2 11 Q13: x 2 6x +5 = ( x 2 6x ) +5 = (x +3) = (x +3) 2 4

13 ANSWERS: UNIT 2 TOPIC 3 9 Q14: 1+6x x 2 = x 2 + 6x + 1 = [ x 2 6x ] +1 [ ] = (x 3) = (x 3) = (x 3) Q15: x = y c m Q16: r = 3 V πh Q17: x = y b m Q18: x = by (a c) + a or x = y b + am m Q19: 2x +3y x +2y Q20: Steps: What is the common denominator? 6h Multiply each term to give it this denominator then simplify. Answer: Q21: Steps: 30 gh 6h What is the common denominator? 6 Multiply each term to give it this denominator then simplify. Answer: 7k +4 6 Q22: b 2 Q23: 2m

National 5 Mathematics Assessment Practice Topic 6: Graphs of quadratic functions and solving quadratic functions

SCHOLAR Study Guide National 5 Mathematics Assessment Practice Topic 6: Graphs of quadratic functions and solving quadratic functions Authored by: Margaret Ferguson Heriot-Watt University Edinburgh EH14