© 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