Download Higher Student Book Ch 20 - Pearson Schools and FE Colleges

20
Quadratic equations
This chapter is about quadratics.
In nature, the growth of a
population of rabbits can be
modelled by a quadratic
equation.
Objectives
This chapter will show you how to
• f actorise quadratic expressions, including the
difference of two squares  B   A
• solve quadratic equations by rearranging  B
• f actorise quadratics and solve quadratic equations of the
form ax2 1 bx 1 c 5 0  B   A   A*
• use the quadratic equations formula  A   A*
• complete the square  A*
Before you start this chapter
Put your calculator away!
1 Factorise
a 3x 1 6y
d 5r2 1 15rt
b 8x 1 x2
e 12xyz 1 6xy 1 18y2 c 3m2 1 mn
2 Work out
a 23 3 4
b 8 3 26 3 21
c 27 3 5 1 12
e (26)2 3 3
7 3 24 1 4
 ​
f​ ___________
    
212
d (2 1 24) 3 23
3 Work out the value of these expressions when
a 5 23, b 5 24 and c 5 2.
____
a abc
b​ √24b ​ 
HELP Chapter 15
d b2 2 4ac c ab2 2 c
HELP
Chapter 12
20.1
L
Factorising the difference
of two squares
Why learn this?
Being able to factorise
a quadratic expression
will help when solving
quadratic equations.
Objectives
Keywords
quadratic expression,
difference of two
squares, factorise
B   A   Factorise a quadratic expression
that is the difference of two squares
Skills check
HELP Section 12.5
1 Expand
a (x 1 3)(x 2 3)
b (x 1 7)(x 2 7)
c (x 2 5)(x 1 5)
What do you notice when you have expanded the brackets?
2 Expand
a (2a 1 4)(2a 2 4) b (3x 2 2)(3x 1 2) c (5m 1 n)(5m 2 n)
Quadratic expressions
A quadratic expression is an algebraic expression whose highest power of x is x2.
They are usually of the form ax2 1 bx 1 c, where a, b and c are numbers and a  0.
These are all quadratic expressions.
3x2 1 2x 1 5 x2 2 3x 2 2 4x2 1 7 12x2 2 3x
These expressions all represent one square number subtracted from another.
x2 2 4 c2 2 64 16a2 2 25
An expression of the form x2 2 b2, where x and b are numbers or algebraic terms, is called the
difference of two squares.
In general, x2 2 b2 5 (x 2 b)(x 1 b). B
Check this by multiplying out (x 2 b)(x 1 b).
Example 1
Factorise x2 2 9.
Remember that factorising is the inverse of expanding brackets.
x2 2 95 x2 2 32
Write as ‘letter squared’ 2 ‘number squared’.
Use x2 2 b2 5 (x 2 b)(x 1 b) with b 5 3.
5 (x 2 3)(x 1 3)
Exercise 20A
B
B
1 Factorise
a x2 2 16
e a2 2 1
b x2 2 25
f m2 2 64
c x2 2 100
g n2 2 36
2 Joe thinks of a number, squares it and subtracts 16.
a Write down an algebraic expression to illustrate this.
AO 2
b Factorise your answer to part a.
298
Quadratic equations
d x2 2 144
h t2 2 121
Example 2
A
Factorise 16m2 2 49.
Notice that 16m2 5 (4m)2.
16m2 2 495 (4m)2 2 72
5 (4m 2 7)(4m 1 7)
Exercise 20B
1 Factorise
a 4x2 2 25
b 9a2 2 36
c 16m2 2 1
d 100t 2 121
e 169z 2 4
f 225q 2 144
2
2
2 Copy and complete. 2
72m 2 50 5 2( 2 )
2 )( 5 2( 1 2
A
Take out 2 as a common factor.
)
3 Factorise each expression by first taking out a common factor.
20.2
L
a 50x2 2 200
b 27m2 2 3
c 80t2 2 45
d 2a2 2 18b2
e 3h2 2 75k2
f 600x2 2 6y2
Factorising quadratics of the
form x2 1 bx 1 c
Why learn this?
Understanding how an algebraic
expression is constructed can
tell you much more about the
expression.
Objectives
Keywords
product, sum,
coefficient
B   Factorise a quadratic expression
of the form x2 1 bx 1 c
Skills check
1 Find two positive numbers whose
a product is 12 and sum is 7
c product is 12 and sum is 13
e product is 212 and sum is 21
b product is 20 and sum is 12
d product is 212 and sum is 1
f product is 212 and sum is 24.
Factorising quadratics of the form x2 1 bx 1 c
Expanding a product of two expressions, like (x 1 2) and (x 1 3), gives a
quadratic expression.
(x 1 2)(x 1 3) 5 x2 1 5x 1 6
20.2 Factorising quadratics of the form x2 1 bx 1 c
299
Factorising is the inverse of expanding.
To factorise a quadratic, you need to write it as the product of two expressions.
5 is the sum of 2 and 3.
x2 1 5x 1 65 x2 1 2x 1 3x 1 6
5 (x 1 2)(x 1 3)
6 is the product of 2 and 3.
In general, to factorise the equation x2 1 bx 1 c, find two
numbers whose sum is b (the coefficient of x) and
whose product is c.
B
The coefficient of x is the
number multiplying the x.
Example 3
Factorise x2 1 7x 1 10.
x2 1 7x 1 10 5 (x 1 5)(x 1 2)
First look for the product, then test the sums.
Find two numbers whose product is
10 and whose sum is 7.
The pairs of numbers whose product
is 10 are
1 and 10 1 1 10 5 11 7
2 and 5
2 1 5 5 7 3
The numbers must be 2 and 5.
Exercise 20C
B
B
1 Factorise each quadratic expression.
a x2 1 5x 1 6
b x2 1 6x 1 8
c z2 1 6z 1 5
d a2 1 11a 1 10
e n2 1 8n 1 15
f f 2 1 12f 1 36
g m2 1 8m 1 12
h x2 1 14x 1 24
i b2 1 11b 1 30
Example 4
Factorise x2 2 7x 1 10.
x2 2 7x 1 10 5 (x 2 2)(x 2 5)
Find two numbers whose product is
10 and whose sum is 27.
The pairs of numbers whose product
is 10 are
1 and 10
1 1 10 5 11 7
2 and 5
2 1 5 5 7 7
21 and 210 21 1 210 5 211 7
22 and 25 22 1 25 5 27 3
The numbers must be 22 and 25.
Exercise 20D
B
300
1 Factorise each quadratic expression.
a x2 2 5x 1 6
b x2 2 9x 1 8
c z2 2 7z 1 12
d a2 2 9a 1 18
e n2 2 10n 1 25
f f 2 2 8f 1 16
g x2 2 13x 1 30
h b2 2 11b 1 28
i p2 2 10p 1 24
Quadratic equations
Example 5
B
Factorise a x2 2 6x 2 7 b x2 1 x 2 12
a x2 2 6x 2 7 5 (x 2 7)(x 1 1)
Find two numbers whose product
is 27 and whose sum is 26.
The pairs of numbers whose
product is 27 are
21 and 7 21 1 7 5 6 7
1 and 27 1 1 27 5 26 3
The numbers must be 1 and 27.
b x2 1 x 2 12 5 (x 2 3)(x 1 4)
Find two numbers whose product is 212 and
whose sum is 1.
The pairs of numbers whose product is 212 are
21 and 12
21 1 12 5 11 7
1 and 212
1 1 212 5 211 7
22 and 6
22 1 6 5 4 7
2 and 26
2 1 26 5 24 7
23 and 4
23 1 4 5 1 3
3 and 24
3 1 24 5 21 7
The numbers must be 23 and 4.
With practice, you will become better
at spotting the correct combination.
General rules for factorising quadratics
In general
• if c is positive and b is positive, both numbers in the brackets will be positive
• if c is positive and b is negative, both numbers in the brackets will be negative
• if c is negative, one number will be negative, one will be positive.
Exercise 20E
1 Factorise
a x2 1 4x 2 12
b x2 2 x 2 20
c z2 2 2z 2 15
d a 1 6a 2 7
e n 1 6n 2 16
f f 2 f 2 30
g m 1 m 2 30
h t 2 6t 2 72
i y2 1 19y 2 120
2
2
2
2
2
B
2 Copy and complete these statements.
a t2 1 7r 2 b m2 2 1 15 5 (t c q 2 12q 2
5 (t 1 10)(t 2 5 (q )
)(t 2 5)
)(q 2 2)
3 a Factorise each expression. Simplify your answers as much as possible.
i x2 1 6x 1 9
iv x2 2 14x 1 49
ii x2 2 8x 1 16 v x2 2 10x 1 25
iii x2 1 4x 1 4
vi x2 1 16x 1 64
b What do you notice about all the answers to part a?
c Copy and complete these statements, where m and n are numbers.
i (x 1 m)2 5 x2 1 x 1 ii (x 2 n)2 5 x2 2 x 1 20.2 Factorising quadratics of the form x2 1 bx 1 c
301
20.3
Solving quadratic equations
Why learn this?
The path of a cricket
ball can be modelled
using a quadratic
equation.
Objectives
B   Solve quadratic equations by rearranging
B   Solve quadratic equations by factorising
Skills check
HELP Section 14.2
1 Solve the equation 2x 1 5 5 10x 2 19.
2 Angel has x CDs in her collection. Write an algebraic expression
for the number of CDs that each of these friends has.
a Amy who has twice as many as Angel.
b Judith who has four less than Amy.
HELP Section 15.1
c Angela who has half as many as Judith.
d Jo who has four times as many as Angela.
Solving quadratic equations by rearranging
You can solve some quadratic equations by rearranging them to make x the subject.
B
Example 6
Solve the quadratic equation 3x2 2 27 5 0.
3x2 2 275 0
3x25 27
x25 9
x5 63
Add 27 to both sides of the equation.
Divide both sides by 3.
Find the square root of both sides.
Remember, when you find the root there
are two solutions: positive and negative.
Exercise 20F
B
1 Solve these equations.
a r2 5 169
d y2 2 20 5 219
b x2 1 5 5 14
m2 ​ 5 6.25
e​ ___
 
4
c 18 5 2t2
p2
 
f​ __ ​ 5 20
5
2 Find the roots of
a 2x2 1 7 5 39
b 5y2 2 100 5 280
c 3r 5 5r 2 98
d 3t 5 t 1 18
e 100 2 5y 5 95
f 2x2 1 2 5 130
2
2
2
302
2
Quadratic equations
2
Root is another
name for a solution.
Keywords
solve, square root,
root
Example 7
B
Solve the equation 2(x 1 3)2 2 5 5 195.
2(x 1 3)2 2 55 195
2(x 1 3)25 200
(x 1 3)25 100
x 1 35 610
Add 5 to both sides of the equation.
Divide both sides by 2.
Take the square root of both sides.
x5 213 or 7
Subtract 3 from both sides.
Remember to give both solutions.
Exercise 20G
1 Find the roots of
a 2(x 1 1)2 5 8
B
(r 2 7)2
b 4.5 5 ​ _______
 
 ​ 
2
2 Solve these equations.
a (x 1 1)2 2 16 5 20
b 100 5 4t2 1 36
c 150 2 3t2 5 42
d 6 1 3t2 5 2t2 1 15
e 4(x 1 3)2 5 100
f 7(y 2 2)2 5 700
3 A field is three times as long as it is wide.
a Using x for the width of the field, write an expression for its length.
b Write an expression for the area of the field, in terms of x.
B
Use your answer to part a.
c The field has an area of 1200 m2.
Write an equation for the area of the field.
Use your answer to part b.
d Solve your equation to find x.
AO 2
e What are the length and the width of the field?
B
4 Explain why you cannot find a solution to x2 1 20 5 5.
5 A rectangle has length five times its width.
The area of the rectangle is 845 mm2.
What is the width of the rectangle?
AO 3
Solving quadratic equations by factorising
Solving quadratic equations by factorising relies on the fact that when a 3 b 5 0, a is 0, b is 0
or both are 0.
So if (x 1 2)(x 2 4) 5 0, either x 1 2 5 0, which means x 5 22, or x 2 4 5 0, which means x 5 4.
If the product of two things is
zero, one of them must be zero.
To solve a quadratic equation
Step 1: Rearrange the equation so that one side is zero.
Step 2: Factorise the quadratic expression.
Step 3: Find the solutions.
Usually there are two solutions.
However, when the expression
factorises to (x 1 m)2 5 0,
there is only one solution.
20.3 Solving quadratic equations
303
B
Example 8
Solve the equation x2 5 3x.
Step 1: x2 2 3x 5 0
Subtract 3x from both sides to make one side zero.
Step 2: x(x 2 3) 5 0
Factorise the expression.
Step 3: x 5 0 or x 2 3 5 0
So x 5 0 or x 5 3
Exercise 20H
B
Solve the equation. If the product of
two numbers is zero, at least one of
the numbers must be zero.
Factorise first.
1 Solve these equations.
a x2 1 7x 5 0
b t2 2 5t 5 0
c 3x2 1 6x 5 0
d y2 5 5y
e 0 5 4w2 2 12w
f 5y 5 20y2
g a 2 a2 5 0
h 5t 5 30t2
i 14r 5 63r2
a 2x2 2 8x 5 0
b 4t2 1 t 5 0
c 7m2 5 14m
d 8g2 5 24g
e 15f 5 6f 2
f 35w 5 10w2
2 Solve these equations.
B
Example 9
Find the roots of the equation x2 2 x 2 8 5 4.
Step 1: x2 2 x 2 12 5 0
Root is another name for a solution.
Step 2: (x 2 4)(x 1 3) 5 0
Subtract 4 from both sides
to make one side zero.
Step 3: x 2 4 5 0 or x 1 3 5 0
Factorise the expression.
So x 5 4 or x 5 23
Exercise 20I
B
B
1 Find the roots of these equations.
a x2 1 4x 1 3 5 0
b x2 2 x 2 6 5 0
c x2 2 6x 1 8 5 0
d x2 1 x 5 12
e x2 5 x 1 20
f x2 1 2x 5 21
g z2 5 3z 1 4
h 2q 1 q2 5 15
i w2 5 4w 2 4
j 6t 1 7 5 t2
k 6p 1 9 5 2p2
l 10x 2 25 5 x2
2 Jane is three years younger than her older sister. The product
of their ages is 54. Use x to represent Jane’s age.
a Write down an algebraic expression for her sister’s age.
Remember that
Jane is younger.
b Write down and simplify an algebraic expression for the product of their ages.
AO 2
304
c Form and solve an algebraic equation to find the value of x.
d Explain why only one of the solutions makes sense.
Quadratic equations
B
3 The height of the rectangle is 3 cm more than the width. a Write down an algebraic expression for the height of
the rectangle.
b Write down an algebraic expression for the area of
the rectangle.
c Given that the area of the rectangle is 40 cm2, form and
solve a quadratic equation to work out the value of t.
t cm
4 A rectangular garden is 4 m longer than it is wide. Its area is 165 m2.
a Sketch and label a diagram to show the area.
b Form and solve a quadratic equation to work out the dimensions of the garden.
5 I think of a negative number, square it and add five times the original number.
My answer is 24. What number did I think of?
AO 2
B
6 I think of a positive number.
I square it, then subtract six times the number.
The answer is 27.
What was my original number?
20.4
L
AO 3
Factorising quadratics of the form
ax 2 1 bx 1 c
Why learn this?
By breaking down an algebraic
expression you can discover
some of the properties of
the expression.
Objectives
A   Solve quadratic equations by factorising
*
A   Factorise quadratic expressions of the form
ax2 1 bx 1 c
Skills check
1 Write down all the pairs of numbers whose product is Don’t forget negative numbers.
a 10
b 12
c 236
2 Factorise
b x2 1 5x 1 6
c x2 2 8x 1 7
a x2 1 3x 2 4
Factorising quadratics of the form ax2 1 bx 1 c
In the expression ax2 1 bx 1 c, the a is the coefficient of x2 and b is the coefficient of x.
In the quadratic expression 3x2 1 13x 1 4, the coefficient of x2 is 3. The first terms in the
brackets must multiply to give 3x2.
The first terms in the brackets must be 3x and 1x. 3 is a prime number – the
only factors are 3 and 1.
3x2 1 13x 1 4 5 (3x )(x )
The product of the last two terms must be 14.
Possible pairs of numbers are 1 and 4, 21 and 24, 2 and 2, or 22 and 22.
The coefficient of x is positive (113), so the two numbers must be positive.
20.4 Factorising quadratics of the form ax2 1 bx 1 c
305
Possible factorisations are
(3x 1 4)(x 1 1) (3x 1 1)(x 1 4) (3x 1 2)(x 1 2)
Try expanding each one.
(3x 1 4)(x 1 1) 5 3x2 1 3x 1 4x 1 4 7
(3x 1 1)(x 1 4) 5 3x2 1 12x 1 x 1 4 3 (3x 1 2)(x 1 2) 5 3x2 1 6x 1 2x 1 4 7
Always check using FOIL
to expand the brackets.
So 3x2 1 13x 1 4 5 (3x 1 1)(x 1 4)
A
Example 10
Factorise 2x2 2 7x 2 4.
2x2 2 7x 2 4 5 (2x )(x )
Pairs of numbers whose product is 24 are
21 and 4, 1 and 24, or 2 and 22.
So the possible factorisations are
(2x 2 1)(x 1 4) 5 2x2 1 8x 2 x 2 4
(2x 1 4)(x 2 1) 5 2x2 2 2x 1 4x 2 4
(2x 1 1)(x 2 4) 5 2x2 2 8x 1 x 2 4
(2x 2 4)(x 1 1) 5 2x2 1 2x 2 4x 2 4
(2x 1 2)(x 2 2) 5 2x2 2 4x 1 2x 2 4
(2x 2 2)(x 1 2) 5 2x2 1 4x 2 2x 2 4
Therefore 2x2 2 7x 2 4 5 (2x 1 1)(x 2 4).
The only factors of 2
are 1 and 2.
One must be positive
and one negative
since the number
term is negative.
This gives the 27x
required. Always
check that the x term
is correct.
With more practice you
will not need to write
out all the combinations
but will be able to work
them out in your head.
Exercise 20J
A
Factorise each quadratic expression.
1 2x2 1 5x 1 3
2 3x2 1 14x 1 8
3 5x2 1 12x 1 4
4 7x2 1 26x 1 15
5 5x2 1 19x 2 4
6 3x2 2 4x 2 4
7 11x2 2 13x 1 2 8 2x2 2 5x 1 2
9 3x2 2 19x 1 20
10 5x2 2 39x 2 8
11 2x2 2 14x 1 24
12 7x2 2 8x 2 12
Factorising quadratics of the form ax2 1 bx 1 c
when the coefficient of x2 is not prime
When the coefficient of x2 is not prime, there are more possible cases to consider.
306
Quadratic equations
Example 11
A
Factorise 6x2 1 11x 1 4.
Factors of 6 are
1 and 6 or 2 and 3.
(3x )(2x ) or (6x )(x )
Pairs of numbers whose product is 14
are 2 and 2 or 1 and 4.
All terms are positive,
so only consider
positive numbers.
So the possible factorisations are
(3x 1 2)(2x 1 2) 5 … 1 6x 1 4x 1 … 5 … 1 10x 1 … (6x 1 2)(x 1 2) 5 … 1 12x 1 2x 1 … 5 … 1 14x 1 … (3x 1 1)(2x 1 4) 5 … 1 12x 1 2x 1 … 5 … 1 14x 1 … (3x 1 4)(2x 1 1) 5 … 1 3x 1 8x 1 … 5 … 1 11x 1 … This gives 111x required.
(6x 1 1)(x 1 4) 5
(6x 1 4)(x 1 1) 5
Therefore 6x2 1 11x 1 4 5 (3x 1 4)(2x 1 1).
You can stop trying
once you have found
the correct pair.
Exercise 20K
1 Factorise each quadratic expression.
a 8x2 1 17x 1 2
b 4x2 1 8x 1 3
c 6x2 1 17x 1 5
d 6x 1 10x 1 4
e 8x 1 20x 1 12
f 30x 1 52x 1 16
2
2
2
A
2 Factorise each quadratic expression.
You need to divide through by a common factor first.
a 2x2 1 8x 1 6 5 2(x2 1 1 b 3x2 1 21x 1 30
)
c 18x2 1 69x 1 60
Example 12
A
Factorise 8x2 2 29x 2 12.
8x2 2 29x 2 12 5 (8x )(x )
or (4x )(2x )
AO 2
Pairs of numbers whose product is 212
are 212 and 1, 12 and 21, 26 and 2, 6 and 22, 24 and 3, 4 and 23.
Possible factorisations are
(4x 2 12)(2x 1 1) 5 8x2 2 20x 2 12
(8x 2 1)(x 1 12) 5 8x2 1 95x 2 12
(8x 1 6)(x 2 2) 5 8x2 2 10x 2 12
There are many possible
combinations. Try different
ones until you find which
one will give you 229x.
The correct factorisation is (8x 1 3)(x 2 4) 5 8x2 2 29x 2 12.
20.4 Factorising quadratics of the form ax2 1 bx 1 c
307
Exercise 20L
A
1 Factorise each quadratic expression.
a 10x2 1 x 2 3
b 12x2 2 x 2 6
d 20x2 1 19x 2 28
e 30x2 2 52x 1 16
c 4x2 1 2x 2 6
2 Factorise each quadratic expression. a 5x 1 5x 2 10
b 14x 1 35x 2 84
c 15x 2 72x 2 15
d 28x2 2 88x 1 12
e 24x2 1 4x 2 4
f 50x2 2 70x 2 60
2
2
A
You need to divide through
by a common factor first.
2
g 36x2 2 42x 1 12
Example 13
Find the values of x which satisfy the equation 8x2 5 14x 1 4.
Step 1: 8x2 2 14x 2 4 5 0
Rearrange the equation to make one side zero.
Step 2: (2x 2 4)(4x 1 1) 5 0
Step 3: 2x 2 4 5 0 or 4x 1 1 5 0
2x 5 4 or 4x 5 21
1
So x 5 2 or x 5 2 ​ __  ​
4
Factorise the equation.
Solve the two linear equations.
Exercise 20M
A
1 Find the roots of these quadratic equations.
Leave your answers as fractions where necessary.
a 2a2 1 5a 2 3 5 0
b 3b2 1 5b 1 2 5 0
c 4c2 2 c 2 5 5 0
d 0 5 5d2 2 8d 2 4
e 6e2 2 16e 1 8 5 0
f 4f 2 2 6f 2 4 5 0
g 6g2 1 19g 1 10 5 0
h 0 5 4h2 1 8h 1 4
i 7i2 2 3i 2 4 5 0
2 Find the values of x which satisfy these equations.
A
AO 2
a 2x2 5 4x 1 6
b 9x2 1 10 5 21x
c 10x2 1 13x 5 9
d 4x 1 16 2 6x2 5 0
e 15x2 5 230x 2 15
f (x 1 2)(x 2 2) 5 3x
3 a Write down an algebraic expression for the area of the rectangle.
b The area of the rectangle is 108 cm2.
Form and solve an algebraic equation to
find the value of x.
2x � 1
c What is the perimeter of the rectangle?
3x
A
4 I think of a number.
Three times the square of my number is equal to twelve times my number.
Work out the possible values of my number.
AO 3
308
Quadratic equations
5 Next year Yvette will be four times her daughter Amelia’s age.
Let x represent Amelia’s age next year.
a Write down an algebraic expression for
i Yvette’s age next year ii Amelia’s age this year iii Yvette’s age this year.
b The product of their ages is 351.
Form and solve an algebraic equation to work out Amelia’s age.
c How old is Yvette this year?
20.5
L
Using the quadratic formula
A*
AO 2
Keywords
quadratic formula,
Objectives
discriminant
A   A*   Solve quadratic equations by
using the quadratic formula
A*   Decide how many solutions a quadratic equation has by
considering the discriminant
Why learn this?
This method solves
quadratic equations
that you can’t factorise,
like x2 1 3x 2 7.
Skills check
There will be
__
two solutions.
3y
1 Using the formula x 5 ​ ___ ​ 2 ​ 
  √z ​ , find the value of x when 8
a y 5 16, z 5 100 b y 5 24, z 5 49
c y 5 80, z 5 100
The quadratic formula
Sometimes a quadratic expression cannot be factorised.
You can use the quadratic formula to solve a quadratic
equation of the form ax2 1 bx 1 c 5 0, where a  0.
________
You do not need to learn the
formula – it will be on the
exam formula sheet.
Be careful! You cannot use the quadratic formula
until you have made one side of the equation zero.
2b 6 ​ √b2 2 4ac ​ 
 ​     
x 5 ​ _______________
2a
A
Example 14
Use the quadratic formula to solve the equation x2 1 3x 2 7 5 0.
a 5 1, b 5 3, c 5 27
Write down the values of a, b and c.
_________
2b 6 ​ √b2 2 4ac ​ 
_______________
x5 ​ 
    
 ​
2a
_________________
  
23 6 ​ √ 32 2 4 3 1 3 27 ​
_________________________
   
  
 ​
2 3 1
5 ​ 
________
23 6 ​ √ 9 1 28 ​ 
    
5 ​ _______________
 ​
2
___
23 6 ​ √ 37 ​ 
___________
Substitute these values into the
quadratic formula. Be careful
with the negative value.
Simplify the calculation. Follow
the order of operations.
    
 ​
2 ___
___
√
√
37 ​
 
37 ​ 
23 1 ​ 
23 2 ​ 
    
    
x5 ​ ___________
 ​
or x 5 ​ ___________
 ​
2
2
5 ​ 
Leave your answer
in surd form.
20.5 Using the quadratic formula
309
Exercise 20N
A
A
Use the quadratic formula to solve each equation. Leave your answers in surd form.
1 x2 1 3x 2 9 5 0
2 x2 1 5x 2 12 5 0
3 x2 1 6x 1 5 5 0
4 x2 1 6x 1 2 5 0
5 3x2 2 2x 2 8 5 0
6 2x2 1 8x 2 20 5 0
7 5y2 1 12y 2 4 5 0
8 12r2 2 8r 1 1 5 0
9 7t2 2 2t 2 8 5 0
10 3g2 1 7g 1 3g 5 0
Example 15
Solve the quadratic equation 2x2 5 6x 1 12.
2x2 2 6x 2 12 5 0
a 5 2, b 5 26, c 5 212
First rearrange the equation
to make one side zero.
_________
2b 6 ​ √b2 2 4ac ​ 
x5 ​ _______________
    
 ​
2a _______________________
   
2 (26) 6 ​ √(26)2 2 4 3 2 3 (212) ​
    
   
 ​
5 ​ _________________________________
2 3 2
_________
6 6 ​ √ 36 1 96 ​ 
    
5 ​ _______________
 ​
4
_____
6 6 ​ √132 ​ 
    
5 ​ __________
 ​
4 ___
6 6 2​ √33 ​ 
    
5 ​ __________
 ​
4___
3 6 ​ √33 ​ 
 
5 ​ _________
 ​ 
2
Exercise 20O
A
A
1 Use the quadratic formula to solve these equations.
Leave your answers in surd form.
Be very careful with positive
and negative numbers.
____
______
3 33 ​ 
5
√
​  132 ​ 5 √
​  4
___
​  33 ​ 
2√
Divide all terms by 2.
Make sure you rearrange
the equations first.
a x2 5 4x 1 1
b x2 1 16 5 12x
c x2 2 8x 5 6
d x2 5 1 2 6x
e 4 1 2x 5 x2
f x2 1 8x 1 2 5 0
2 Try to solve this quadratic equation using the quadratic formula.
x2 1 x 1 1 5 0
Explain why you cannot find a solution.
AO 2
A*
3 A carpet manufacturer wishes to make carpet tiles with area 1500 cm2.
AO 3
4 Look at your answer to Q3. Did you need to use the quadratic formula?
310
The tiles are rectangular and the length is 10 cm less than double the width.
Work out the dimensions of a carpet tile.
Explain your answer.
Quadratic equations
The discriminant
Question 2 in Exercise 20O asked you to try to solve the equation x2 1 x 1 1 5 0.
Using the quadratic formula
a 5 1, b 5 1, c 5 1
________
2b 6 ​ √b2 2 4ac ​ 
 ​
    
x5 ​ _______________
2a
______________
21 6 ​ √12 2 4 3 1 3 1 ​
  
 ​
5 ​ _____________________
   
  
2 3 1
___
21 6 ​ √23 ​ 
5 ​ __________
 ​ 
 
2
The calculations result in trying to find the square root of a negative number. This has no real
solutions – you will learn more about this if you do A-level maths.
b2 2 4ac in the quadratic formula is known as the discriminant.
In general,
• when b2 2 4ac . 0, there are two distinct solutions to the quadratic equation
• when b2 2 4ac , 0, there are no real solutions to the quadratic equation
• when b2 2 4ac 5 0, there is one solution (sometimes called a repeated root).
Example 16
By considering the discriminant, decide whether each of these quadratic equations
has zero, one or two solutions.
A*
a 3x2 1 2x 2 5 5 0
b 7x2 5 10x 2 8
c 9x2 1 16 5 24x
a 3x2 1 2x 2 5 5 0
Write down the values of a, b and c.
a 5 3, b 5 2, c 5 25
Work out b2 2 4ac
b2 2 4ac5 22 2 4 3 3 3 25
5 4 1 60
5 64
Since 64 . 0 there are two solutions.
b 7x2 5 10x 2 8
Rearrange to the form ax2 1 bx 1 c 5 0
7x2 2 10x 1 8 5 0
a 5 7, b 5 210, c 5 8
b2 2 4ac5 (–10)2 2 4 3 7 3 8
5 100 2 224
5 2124
Since 2124 , 0 there are no solutions.
20.5 Using the quadratic formula
311
c 9x2 1 16 5 24x
Rearrange to the form ax2 1 bx 1 c 5 0
9x2 2 24x 1 16 5 0
a 5 9, b 5 224, c 5 16
b2 2 4ac5 (224)2 2 4 3 9 3 16
5 576 2 576
5 0
There is one (repeated) solution.
Exercise 20P
A*
1 For each quadratic equation, decide if there are zero, one or two solutions.
a 3x2 1 2x 2 4 5 0
b 5m2 1 9m 1 6 5 0
c 3t2 1 6t 1 3 5 0
d 4d2 2 5d 1 6 5 0
e 0 5 2z2 1 5z 1 1
f 4x2 5 3x 2 1
g 9t 5 5t2 2 12
h 2q2 5 2 8q 2 8
20.6
L
Completing the square
Why learn this?
In mathematics, as in
life, it is important to
have more than one way
to solve a problem.
Objectives
A*   Solve a quadratic equation by completing
the square
Skills check
HELP
1 Expand and simplify
a (x 1 2)2
b (x 2 3)2
Section 12.5
c (x 2 5)2
Completing the square
Completing the square is another way to solve a quadratic equation which cannot be factorised.
Expanding an expression of the form (x 1 a)2 gives
(x 1 a)(x 1 a) 5 x2 1 2ax 1 a2
Working backwards, this can be used to ‘complete the square’.
Consider the equation x2 1 4x 1 10 5 0.
For the coefficient of x to be 4 the squared bracket must be (x 1 2)2.
But (x 1 2)2 5 x2 1 4x 1 4.
2
2
This is half the coefficient of x.
To get from (x 1 2) to x 1 4x 1 10 you need to subtract 4 and then add 10.
x2 1 4x 1 10 5 0
(x 1 2)2 2 4 1 10 5 0
(x 1 2)2 1 6 5 0
312
Quadratic equations
Example 17
A*
2
Write the expression x 1 10x 1 9 in completed square form.
Halve the coefficient of x.
2
2
(x 1 5) 5 x 1 10x 1 25
The expression required is x2 1 10x 1 9
Subtract the square of the
number in the bracket.
5 (x2 1 10x 1 25) 2 25 1 9
5 (x 1 5)2 2 25 1 9
Put in the original
number term.
5 (x 1 5)2 2 16
Exercise 20Q
1 Write each expression in completed square form.
a x2 1 6x 1 3
b x2 1 2x 1 7
c x2 2 8x 1 5
d x2 2 12x 1 12
e x2 2 4x 2 7
f x2 2 10x 2 1
A*
Be careful with the values of p and q.
Are they positive or negative?
2 Write each algebraic expressions in the form 2
(x 1 p) 1 q, giving the values of p and q.
a x2 1 10x 1 32
b x2 1 2x 1 2
c x2 2 4x 1 20
d x2 2 14x 1 10
e x2 2 6x 2 3
f x2 2 4x 2 2
Example 18
By completing the square, solve the equation x2 2 8x 1 5 5 0.
A*
Leave your answer in surd form.
x2 2 8x 1 55 (x 2 4)2 2 16 1 5
Write the expression in
completed square form.
5 (x 2 4)2 2 11
So (x 2 4)2 2 115 0
(x 2 4)25 11
Solve the equation
by rearranging.
___
x 2 45 6 ​ √11 ​ 
___
x5 4 6 ​ √11 ​ 
This is the exact answer in surd form.
Exercise 20R
1 Solve the quadratic equations by completing the square.
a x2 1 10x 1 9 5 0
b x2 1 2x 2 8 5 0
c x2 2 8x 1 10 5 0
d x2 2 12x 1 16 5 0
e x2 2 4x 2 4 5 0
f x2 1 6x 2 7 5 0
20.6 Completing the square
A*
313
A*
2 Give the exact solution to these quadratic equations
by completing the square.
a x2 1 8x 2 9 5 0
b x2 1 4x 2 8 5 0
c x2 2 2x 2 1 5 0
d x2 2 8x 1 10 5 0
e x2 2 20x 1 50 5 0
f x2 2 14x 1 41 5 0
3 Solve these quadratic equations Leave your answers in surd
form where apprpriate.
Don’t forget to rearrange the equations first.
by completing the square.
a x2 5 6x 2 4
b 3x(x 1 6)5 6
c (x 1 1)(x 2 5) 5 7
d (x 2 2)(x 1 8) 5 7
2   ​  
e​ _____
5 x
x 1 6
3    ​ 5 1
f​ ____________
(r 2 1)(r 1 2)
Review exercise
B
1 Factorise the expression x2 1 6x 1 5.
B
2 I think of a number, square it, then subtract three times the number. The result is 108.
Form and solve an algebraic equation to work out the possible values of
the number I thought of.
[2 marks]
[4 marks]
3 I think of a number, square it, then add it to 5 times the number.
AO 3
A
The answer is 24. Form and solve an algebraic equation to work out the possible
values of the number I thought of. [4 marks]
4 a Factorise 2x2 2 15x 2 8.
[2 marks]
b Hence solve the equation 2x2 2 15x 2 8 5 0.
5 Factorise 6y2 1 13y 2 5.
[2 marks]
[2 marks]
6 Use the quadratic formula to solve 2x2 2 6x 1 1 5 0.
Leave your answer in surd form.
7 a Factorise the quadratic expression 6x2 2 11x 2 10.
b Hence solve the equation 6x 2 11x 2 10 5 0.
Leave your answers as fractions.
[3 marks]
[2 marks]
2
A*
[2 marks]
8 A rectangular piece of land has length 3 m more than double the width.
The area of the rectangle is 170 m2.
Work out the dimensions of the rectangle. [5 marks]
9 A rectangular rug is 6 m longer than its width.
AO 3
314
The area of the rug is 16 m2.
Calculate the dimensions of the rug. Quadratic equations
[5 marks]
10 a Find the values of m and n such that x2 1 4x 2 6 5 (x 1 m)2 2 n.
[2 marks]
b Hence solve the equation
x2 1 4x 2 6 5 0 by rearranging, leaving your answer
__
√
in the form a 6 ​  b ​. 
[3 marks]
A*
11 How many roots does each of these quadratic equations have?
a 5x2 2 2x 2 7 5 0
b 3x2 2 11x 1 12 5 0
c 4x2 2 12x 1 9 5 0
[6 marks]
12 a Write the following algebraic expression in completed square form.
x2 2 4x 1 2
[2 marks]
b Hence find the exact solution to the equation x2 2 4x 1 2 5 0.
[2 marks]
Chapter summary
In this chapter you have learned how to
• factorise a quadratic expression of the form
x2 1 bx 1 c  B
• solve quadratic equations by rearranging  B
• factorise a quadratic expression that is the
difference of two squares  B   A
• solve quadratic equations by
factorising  B   A
• factorise quadratic expressions of the form
ax2 1 bx 1 c  A   A*
• solve quadratic equations by using the
quadratic formula  A   A*
• decide how many solutions a quadratic equation
has by considering the discriminant  A*
• solve a quadratic equation by completing the
square  A*
Chapter 20 Summary
315