Orienta

tion

A6 Solve and use quadratic equations

A7 Drawing parabolas and solving simultaneous equations

■ Solve quadratic equations by factorisation and completing the square

■ Sketch quadratic graphs

A-level

Maths, Engineering, Science,

A1 Factorise quadratic expressions

A3 Solve linear equations

What I need to know What I will learn What this leads to

Quadratic equationsWhen a traffic policeman arrives at the scene of a road accident, they measure the length of the skid marks and assess the road conditions. They can then use a quadratic equation to calculate the speed of the vehicles and hence reconstruct exactly what happened.

What’s the point?Quadratic equations have many applications in science and engineering and have played a central role within the development of mathematics itself.

A4

Quadra

tic equatio

ns

Check in You should be able to

■ factorise quadratic expressions1 Factorise each of these expressions.

a x2 � 5x b x2 � 10x � 21c x2 � 25 d y2 � 6y � 9e p2 � 100 f 3ab � 9a2

g 16m2 � 49 h 3x2 � 7x � 2

■ use quadratic formulae2 Given that s � ut � 1 __ 2 at2, find:

a s when u � 8, t � 5 and a � �4b u when s � 100, t � 4 and a � �2

Ric

h t

ask A man wants to enclose part of his garden into a rectangle. He uses a wall in

his garden as one side of the rectangle, and uses 40 m of fencing to enclose the other three sides.What is the maximum area of rectangle he can enclose?What is the maximum area of rectangle he can enclose for L metres of fencing?

Probability revision

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mA

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Pro

ble

m

Grade A/A*

p.122

p.126

p.202

Exercise A4.1

Spot the x2 term, the equation is quadratic.

If two expressions multiply to give zero, one of them must be zero.

Exa

mple

p Solve x2 � 6x.

x2 � 6x � 0 Rearrange so that one side � zero.x(x � 6) � 0 Factorise.Either x � 0 or (x � 6) � 0Either x � 0 or x � 6

Exa

mple

p Solve a x2 � 5x � 6 � 0 b 4x2 � 81.

a x2 � 5x � 6 � 0 (x � 3)(x � 2) � 0 Either x � 3 � 0 so x � �3 This equation has two solutions. or x � 2 � 0 so x � �2

b 4x2 � 81 4x2 � 81 � 0 Factorise – DOTS. (2x � 9)(2x � 9) � 0 Either 2x � 9 � 0 so 2x � 9 so x � 4 1 _

2

or 2x � 9 � 0 so 2x � �9 so x � �4 1 _ 2

Exa

mple

p Solve 5x2 � 14x � 3 � 0.

5x2 � 14x � 3 � 0(5x � 1)(x � 3) � 0

Either 5x � 1 � 0 or x � 3 � 0 x � 1 _

5 x � �3

● Quadratic equations contain a squared term as the highest power, for example 2x2 � 7x � 9 � 0.

● Quadratic equations can have 0, 1 or 2 solutions.

For example x2 � 100 x2 � 0 x2 � �25 Solution(s) x � 10 or �10 x � 0 Impossible Number of solutions 2 1 0

● Many quadratic equations can be solved by: – rearranging so that one side equals zero – factorising.

1 First factorise these quadratic equations, by using the common factor, then solve them.

a x2 � 3x � 0 b x2 � 8x � 0c 2x2 � 9x � 0 d 3x2 � 9x � 0e x2 � 5x f x2 � 7xg 12x � x2 h 4x � 2x2

i 6x � x2 � 0 j 9y � 3y2 � 0k 0 � 7w � w2

2 Solve these quadratic equations by factorising into double brackets.

a x2 � 7x � 12 � 0 b x2 � 8x � 12 � 0c x2 � 10x � 25 � 0 d x2 � 2x � 15 � 0e x2 � 5x � 14 � 0 f x2 � 4x � 5 � 0g x2 � 5x � 6 � 0 h x2 � 12x � 36 � 0i 2x2 � 7x � 3 � 0 j 3x2 � 7x � 2 � 0k 2x2 � 5x � 2 � 0 l 6y2 � 7y � 2 � 0m x2 � 8x � 12 n 2x2 � 7x � 15o 0 � 5x � 6 � x2 p x(x � 10) � �21

3 Solve these quadratic equations by factorising using the difference of two squares.

a x2 � 16 � 0 b x2 � 64 � 0c y2 � 25 � 0 d 9x2 � 4 � 0e 4y2 � 1 � 0 f x2 � 169g 4x2 � 25 h 36 � 9y2

4 Solve these quadratic equations.

a 3x2 � x � 0 b x2 � 2x � 15 � 0c 3x2 � 11x � 6 � 0 d 9y2 � 16 � 0e 25 � 16x2 f x2 � xg 20x2 � 7x � 3 h 8x � 12 � x2

5 Solve these equations.

a 5 � x � 6x(x � 1) b (x � 1)2 � 2x(x � 2) � 10

c 10x � 1 � 3 __ x d 2 _____

x � 2 � 4 ______ x � 1

� 0

e x4 � 13x2 � 36 � 0

This spread will show you how to:● Solve quadratic equations by factorisation

A4.1 Solving quadratic equations

6 A rectangle has a length that is 7 cm more than its width, w. The area of the rectangle is 60 cm2.a Write an algebraic expression for the area of

the rectangle.b Show that w2 � 7w � 60 � 0.c Find the dimensions of the rectangle.

w

Keywords

DOTSFactoriseQuadratic

Probability revision

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Exa

mple

p Complete the square on x2 � 8x � 12.

x2 � 8x � 12 � (x � 4)2 � q

� (x � 4)2 � 28

Exa

mple

p Write these equations in the form (x � p)2 � q.

a x2 � 20x � 1 b x2 � 5x � 8 c 2x2 � 8x � 6

a x2 � 20x � 1 � (x � 10)2 � q � (x � 10)2 � 1 � 100 � (x � 10)2 � 99

b x2 + 5x + 8 � ( x + 5 _ 2 ) 2 + q

� ( x + 5 _ 2 ) 2 + 8 � ( 5 _

2 ) 2

� ( x + 5 _ 2 ) 2 + 7 _

4

c 2x2 � 8x � 6 � 2(x2 � 4x � 3) � 2[(x � 2)2 � 1] � 2(x � 2)2 � 2

Exercise A4.2

1 Complete the square on these quadratic expressions.

a x2 � 4x � 6 b x2 � 8x � 15c x2 � 10x � 26 d x2 � 4xe x2 � 12x � 10 f x2 � 14x � 25g x2 � 4x � 10 h x2 � 8x � 3i x2 � 16x � 1 j x2 � 3x � 4k x2 � 5x � 6 l x2 � 7x � 10m x2 � 9x n x2 � 5x � 2o x2 � 11x � 4

2 Write these expressions in the form (x � p)2 � q where p and q are positive or negative integers.

a x2 � 30x � 90 b x2 � 2 � 16xc 7x � x2 d x2 � 17x � 2

e x2 + 1 _ 2 x � 1 f x2 � 9x � 11

3 Complete the square on these expressions.

a 2x2 � 6x � 4 b 3x2 � 6x � 9c �x2 � 6x � 2 d 5x2 � 10x � 15e 6x � 8 � x2 f 2x2 � 7x � 3

4 a Explain why it is difficult to make x the subject of x2 � 2bx � c.b Complete the square on x2 � 2bx.c Using your answers from parts a and b, show that it is possible to

make x the subject of x2 � 2bx � c and that

x � � √ ______

c � b2 � b

5 Make x the subject of these equations by completing the square first.

a x2 � 4cx � k b x2 � 6x � t3

c x2 � m � 6gx � 0 d 2x2 � 4cx � p

AO

3P

roble

m 6 By completing the square on ax2 � bx � c � 0, prove the quadratic equation formula.

x � �b � √ ________

b2 � 4ac _______________

2a

● Completing the square means writing a quadratic as a squared bracket plus an extra term.

● Some quadratics factorise into squared brackets without an extra term.

For example, x2 � 2x � 1 � (x � 1)(x � 1) � (x � 1)2

x2 � 4x � 4 � (x � 2)(x � 2) � (x � 2)2

x2 � 6x � 9 � (x � 3)(x � 3) � (x � 3)2

x2 � 8x � 16 � (x � 4)(x � 4) � (x � 4)2

● Some quadratics factorise into the form (x � p)2 � q.

For example,

12

Coefficient of x � 10 � 10 � 5 → (x � 5)(x � 5)

x2� 10x � 15 (x � 5)2� ? (x � 5)2� 10

Expanding (x � 5)(x � 5) givesconstant term 25. You onlywant 15, so subtract 10.

12 � �6 � �3 → (x � 3)2

x2� 6x � 20 (x � 3)2� ? (x � 3)2� 11

(x � 3)2→ constant 9.You want 20, so add 11.

Notice that the number in the squared bracket is half the coefficient of the x term in the original quadratic expression.

(�4)2 is 16, so subtract 28 to give the �12 required.

You can write the constant you want (1) and subtract 100 (102).

This means ‘complete the square’.

With odd coefficients, it may be best to work with fractions.

8 � 25 __ 4 � 32 � 25

______ 4 � 7 __ 4

Factorise first.

This spread will show you how to:● Solve quadratic equations by completing the square

Completing the squareA4.2

Keywords

Coefficient

Probability revision

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Grade A*

1 Solve these quadratic equations by completing the square.a x2 � 12x � 20 � 0 b x2 � 2x � 15 � 0c x2 � 4x � 5 � 0 d x2 � 2x � 1 � 0e x2 � 2x � 63 � 0 f x2 � 14x � 49 � 0g x2 � 8x � 0 h y2 � 1 � 12yi p2 � 3p � 2

2 Use completing the square to explain why x2 � 8x � 17 � 0 has no solutions.

3 Solve these equations by completing the square.a y(y � 3) � 88 b x(x � 2) � 143c x(x � 7) � �21 d 2x2 � 6x � 3 � 0e (w � 1)2 � 2w(w � 2) � 10

4 The area of Painting A is twice the area of Painting B.

10 � x

Painting A

6

2x � 1

x

Painting B

a Show that x2 � 2x � 15 � 0.b By completing the square, find the value of x and the dimensions

of each painting.

5 Use completing the square to explain why x2 � 4x � 10 is never less than 6.

6 Find the coordinates of the minimum point of each of these functions.a y � x2 � 8x � 12 b y � x2 � 10x � 5c y � x2 � 12x � 4 d y � x2 � 3x � 1

e y = 1 ___________ x2 � 6x � 3

Exercise A4.3Exa

mple

p Solve y2 � 10y � 21 � 0 by completing the square.

y2 � 10y � 21 � 0(y � 5)2 � 25 � 21 � 0 (y � 5)2 � 4 � 0 (y � 5)2 � 4Either y � 5 � 2 or y � 5 � �2Hence y � �3 or y � �7

Exa

mple

p Show that, if y � x2 � 8x � 10, then y � �26 for all values of x.

y � x2 � 8x � 10y � (x � 4)2 � 10 � 16y � (x � 4)2 � 26Since (x � 4)2 � 0, y � 0 � 26 ⇒ y � �26

Exa

mple

p Find the co-ordinates of the minimum point on the graph of y � x2 � 4x � 5.

y � x2 � 4x � 5 � (x � 2)2 � 9Minimum value of y � �9, where x � 2. Minimum point is (2, �9).

A positive integer has two square roots.

Don’t forget the two square roots.

Any number squared is �0.

When x � �3, y � x2 � 6x � 14 � 5.On a graph of this quadratic expression, the lowest (minimum) point would be at the point (�3, 5).

Solving quadratics by completing the square

This spread will show you how to:● Solve quadratic equations by completing the square● Use completing the square to find the minimum value

A4.3

● You can use completing the square to solve quadratic equations.

For example: x2 � 8x � 7 � 0 (x � 4)2 � 9 � 0 (x � 4)2 � 9 x � 4 � 3 or x � 4 � �3 x � �1 x � �7● You can use completing the square to find the minimum value that a

quadratic expression can have.

x2 � 6x � 14 � (x � 3)2 � 5

The minimum value of x2 � 6x � 14 is 0 � 5 � 5. This occurs whenx � �3.

This part of the expression is a square. The smallest it can be is zero, when x � �3.

This part of the expression is constant, it is always equal to 5.

(�3, 5)

7 Prove that p2 � 6p � 9 can never be negative.

Keywords

MinimumSolve

230230 231

Unit

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Exercise A4.4 Grade A/A*

230

Sketching quadratic graphsA4.4Exa

mple

p Which quadratic functions cross the x-axis at (�3, 0) and (5, 0)?

x � �3 and x � 5 are the solutions to 0 � (x � 3)(x � 5). So the functions are of the form: f(x) � a(x � 3)(x � 5) � a(x2 � 2x � 15).where a is any non-zero constant.

● You can write a quadratic expression as a function, for example f(x) � x2 � 2.

● The graph of a quadratic function is always a parabola.

f(x)

x

Minimumpoint

f(x) � x2

f(x)

x

Maximumpoint

f(x)� �x2

The graphs are symmetrical.f(a) � a2 � f(�a)

● To sketch a quadratic graph, you should show – where it intersects the axes – the coordinates of its turning point (minimum or maximum point).You can work these out from the quadratic function.For example, for y � f(x) � x2 � 6x � 8

�1

1

2

3

4

5

6

7

8

9

10

�2

�1 10 2 3 4 5 6

y

x

y-axis intercept

On the y-axis, x � 0.Substitute x � 0 into the function:f(0) � 02 � 6 � 0 � 8 � 8y-intercept is (0, 8).x-axis intercept(s)

On the x-axis, y � 0.Substitute y � f(x) � 0 and solve the quadratic:0 � x2 � 6x � 80 � (x � 2)(x � 4)Either x � 2 � 0 so x � 2 or x � 4 � 0 so x � 4The intercepts are (2, 0) and (4, 0).Turning point

Find the minimum value by completing the square.f(x) � x2 � 6x � 8f(x) � (x � 3)2 � 1For the minimum value x � 3 � 0, x � 3 and f(3) � �1.The minimum value or turning point is (3, �1).

You can plot a quadratic function by tabulating (x, y) values.

1 Find the y-intercept of each of these quadratic functions.

a y �x2 � 8x � 12 b f(x) � x2 � 8x � 15

c y � x2 � 6x � 5 d f(x) � x2 � 5x � 6

e y � x2 � 10x � 25

2 Find the coordinates where each of the quadratic graphs in question 1 intercept the x-axis.

3 Find the coordinates of the minimum point of each of the quadratic graphs in question 1.

4 Explain why y � x2 � 8x � 20 does not intersect the x-axis.

5 Sketch a graph of the quadratic function y � x2 � 7x � 12, indicating the coordinates of any points of intersection with the axes and of the minimum point.

6 Match these sketch graphs with the equations given. Sketch the remaining function, labelling its points of intersection with the axes.a

y

x�4 5

b

y

x�6 4

c

y

x4

i y � (x � 4)2

ii y � x2 �x � 20

iii y � (x � 6)(x � 4)

iv y � x2 �2x � 24

7 State the equation of these quadratic functions, using the information given in their sketch graphs.

a

�5

y

x

12(�2 , )3

4�18

b

6

y

x36

c

(4, �2)

12y

x

8 Give the equation of a quadratic function thata intersects the y-axis at (0, 5) but that does not intersect the x-axisb intersects the x-axis just once, at (�12, 0)c has a maximum value of (4, 9).

AO

3P

roble

m 9 a Explain why f(x) � (x � 2)(x � 3)(x � 4) is not a parabola. What shape is it?

b Sketch the graph of f(x) � (x � 2)(x � 3)(x � 4).

Where does f(x) intersect the x-axis?

Keywords

FunctionInterceptMaximumMinimumParabola

This spread will show you how to:● Recognise the shape of a graph of a quadratic function● Generate points and plot graphs of general quadratic functions

232

A4 Summary

Check out

233

Exam questions

Edexcel Limited, June 2008

The probability that it will be red is more than 14.

It is twice as likely to be white as red.

1 a Factorise 2x2 � 7x � 15 (2) b Solve 2x2 � 7x � 15 � 0

2 For all values of x, x2 � 6x � 15 � (x � p)2 � qa Find the value of p and the value of q. (2)b On a copy of the axes, draw a sketch of the graph y � x2 � 6x � 15

y

x0

(2)(Edexcel Limited 2007)

You should now be able to:

● Solve simple quadratic equations by factorisation and completing the square

● Sketch graphs of quadratic functions

(2x � 5)(3x � 2) � 6x2 � 4x � 15x � 10 � 6x2 � 11x � 10

x2 � 6x � 5 � (x � 3)2 � q � (x � 3)2 � 5 � 9 � (x � 3)2 � 14q � �14p � 3

OR

(x � p)2 � q � (x � p)(x � p) � q � x2 � px � px � p2 � q � x2 � 2px � p2 � q2p � 6 and p2 � q � �5 p � 3 and 32 � q � �5 9 � q � �5 q � �14 p � 3

There should be 4 terms.

State the values of p and q.

State the values of p and q.

Worked exam questiona Expand and simplify (2x � 5)(3x � 2) (3)b Given that x2 � 6x � 5 � (x � p)2 � q for all values of x, fi nd the value of i p, ii q. (3) (Edexcel Limited 2006)

a

b

b

AO

3 3 y � x2 � 4x � 5 Give the coordinates of the minimum point on this graph. (4)

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Case study 5

To Come

Case study 5

To Come