SCHOLAR Study Guide
National 5 Mathematics
Assessment Practice
Topic 3:
Expanding brackets,
factorising, completing the square,
changing the subject of a formula
and algebraic fractions
Authored by:
Margaret Ferguson
Heriot-Watt University
Edinburgh EH14 4AS, United Kingdom.
First published 2014 by Heriot-Watt University.
This edition published in 2016 by Heriot-Watt University SCHOLAR.
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SCHOLAR Study Guide Assessment Practice Topic 3: National 5 Mathematics
1. National 5 Mathematics Course Code: C747 75
Acknowledgements
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created these materials, and to the many colleagues who reviewed the content.
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Grateful acknowledgement is made for permission to use the following material in the
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The content of this Study Guide is aligned to the Scottish Qualifications Authority (SQA)
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1
Topic 1
Expanding brackets, factorising,
completing the square, changing
the subject of a formula and
algebraic fractions
Contents
3.1
Learning points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
3.2
Assessment practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2
TOPIC 1. 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.
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.
© H ERIOT-WATT U NIVERSITY
TOPIC 1.
EXPANDING BRACKETS, FACTORISING, COMPLETING THE SQUARE,
CHANGING THE SUBJECT OF A FORMULA AND ALGEBRAIC FRACTIONS
1.1
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 − k2 + c OR −[(x + k)2 − k2 ] + 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.
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3
4
TOPIC 1. 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
x
3
1
x
1
2 = 1 and x = 1, 6 = 2 and x2 = x .
Simplifying algebraic fractions may require factorising first.
Before adding and subtracting algebraic fractions you must have a common denominator
e.g.
2y+3x
2
3
x + y = xy
Multiplying fractions is the simplest, you multiply the numerators together then multiply
the denominators together
2x
x
2x2
3y × 5 = 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
x
4x
3
12x
12
y ÷ 3 = y × x = xy = y
Simplify the answer if necessary.
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TOPIC 1.
EXPANDING BRACKETS, FACTORISING, COMPLETING THE SQUARE,
CHANGING THE SUBJECT OF A FORMULA AND ALGEBRAIC FRACTIONS
1.2
5
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
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)(x2 − 2x + 5)
..........................................
Factorising
Factorise these equations.
Q6: 6k 2 − 9k
..........................................
Q7: j 2 − 81
..........................................
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Go online
6
TOPIC 1. EXPANDING BRACKETS, FACTORISING, COMPLETING THE SQUARE,
CHANGING THE SUBJECT OF A FORMULA AND ALGEBRAIC FRACTIONS
Q8:
25m2 − 36n2
..........................................
Q9:
x2 + 4x − 12
..........................................
Q10: 20x2 − 80
..........................................
Q11: 2x2 − x − 10
..........................................
Completing the square
Q12: Express x2 + 6x − 2 in the form (x + p)2 + q
..........................................
Q13: Express x2 − 6x + 5 in the form (x − p)2 + q
..........................................
Q14: Express 1 + 6x − x2 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.
..........................................
Q16: The formula for the volume of a cone is V =
formula.
1
3
π r 2 h, make r the subject of the
..........................................
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)
..........................................
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TOPIC 1.
EXPANDING BRACKETS, FACTORISING, COMPLETING THE SQUARE,
CHANGING THE SUBJECT OF A FORMULA AND ALGEBRAIC FRACTIONS
Q20:
5
h
−
g
6
..........................................
Q21:
2k + 5
3
k−2
2
+
..........................................
Q22:
b4
7
×
14
4b3
..........................................
Q23:
4 m2
n
÷
2m
n
..........................................
..........................................
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7
8
ANSWERS: TOPIC 3
Answers to questions and activities
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:
5c2 − 15cd
Q2:
y 2 + 4y − 21
Q3:
6a2 − a − 2
Q4:
2g 2 + 8g − 10
Q5:
Steps:
•
What is 3x(x2 - 2x + 5)? 3x3 - 6x2 + 15x
•
What is 1(x2 - 2x + 5)? x2 - 2x + 5
•
Collect like terms from these answers to simplify.
Answer: 3x3 − 5x2 + 13x + 5
Q6:
3k(2k − 3)
Q7:
(j − 9)(j + 9)
Q8:
(5m − 6n)(5m + 6n)
Q9:
(x − 2)(x + 6)
Q10:
Hint:
•
Find the highest common factor first.
Answer:20(x − 2)(x + 2)
Q11: (2x − 5)(x + 2)
Q12:
x2 + 6x − 2 = x2 + 6x
− 2
2
= (x + 3) − 9 − 2
= (x + 3)2 − 11
Q13:
x2 − 6x + 5 = x2 − 6x
+ 5
= (x + 3)2 − 9 + 5
= (x + 3)2 − 4
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ANSWERS: TOPIC 3
9
Q14:
1 + 6x − x2 = −x2 + 6x + 1
+ 1
= − x2 − 6x
= − (x − 3)2 − 9 + 1
= −(x − 3)2 + 9
+ 1
2
= −(x − 3) + 10
Q15: x =
Q16: r =
y−c
m
3V
πh
Q17: x =
y−b
m
Q18: x =
by
(a − c)
Q19:
+ a or x =
y −b+am
m
2x + 3y
x + 2y
Q20:
Steps:
•
What is the common denominator? 6h
•
Multiply each term to give it this denominator then simplify.
Answer:
30 − g h
6h
Q21:
Steps:
•
What is the common denominator? 6
•
Multiply each term to give it this denominator then simplify.
Answer:
Q22:
7k + 4
6
b
2
Q23: 2m
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