© Nuffield Foundation 2011

Nuffield Free-Standing Mathematics Activity

Completing the square

means writing the unknown terms of a quadratic in a square bracket

Completing the square

33 xx23

xbecause

9332 xxx

23 2

x

962 xxApplicationTo find the maximum or minimum value of this function.

Example 762 xx

Think about…What is always true about the value of (x + 3)2?What will always be true for (x + 3)2 – 2?

762 xxMinimum value of is –2

Application

To find the minimum point on the graph of the function.

has a minimum value when x = –3 23 2

x762 xx

762 xxyMinimum point on the curve is (–3 , –2)

Think about… For what value of x is (x + 3)2 – 2 a minimum?

Think about… What shape is the graph of y = x2 + 6x + 7 ?Can completing the square help you sketch the graph of a quadratic?

0

y

x

7

(–3, –2)

line of symmetryx = –3762 xxy

Minimum point is (–3 , –2)

23 2

x

When x = 0, y = 7

x = –3 is a line of symmetry

So the intercept on the y axis is 7

Graph of y = x2 + 6x + 7

0

y

x

5

11 xx

122 xx

has a minimum value when x = 1

Example542 2 xx

Minimum point is (1, 3)

312 2

x

522 2

xx

21

x

5112 2

x

To sketch the graph of 542 2 xxy

542 2 xxy 312 2

x

Intercept on the y axis is 5 (1, 3)

line of symmetry

x = 1

542 2 xxy

Think about… What is the maximum or minimum value?

Think about…What shape will the graph be?Where is its turning point?

22 xx

442 xx

has a maximum value when x = –2

Example243 xx

Maximum point is (–2, 7)

227

x

0

y

x

3

xx 43 222

x

423 2x

To sketch the graph of243 xxy 243 xxy 227

x

Intercept on the y axis

is 3

(–2, 7)

x = –2

243 xxy

Think about… What is the maximum or minimum value?

Think about… What shape will the graph be?Where is its turning point?

Note you can find the intercepts on the x axis by solving

043 2 xx

Reflect on your work

How does completing the square help you to sketch the graph of the function?

Can you use completing the square to tell you whether the quadratic function has any real roots?

Completing the square