1. Nuffield Free-Standing Mathematics Activity
Completing the square

2. Completing the square
means writing the unknown terms of a quadratic in a square bracket
Example
because
Application
To find the maximum or minimum value of this function.
Think about… What is always true about the value of (x + 3)2? What will always be true for (x + 3)2 – 2?
Minimum value of
is –2

3. Application
To find the minimum point on the graph of the function.
Think about… For what value of x is (x + 3)2 – 2 a minimum?
has a minimum value
when x = –3
Minimum point on the curve
is (–3 , –2)
Think about… What shape is the graph of y = x2 + 6x + 7 ?
Can completing the square help you sketch the graph of a quadratic?

4. Graph of y = x2 + 6x + 7
line of symmetry y x
x = –3
Minimum point 7
is (–3 , –2)
x = –3 is a line of symmetry
When x = 0, y = 7
(–3, –2)
So the intercept on the y axis is 7

5. Example
Think about… What is the maximum or minimum value?
Think about… What shape will the graph be?
Where is its turning point?
To sketch the graph of
line of symmetry y x
has a minimum value
when x = 1 5
Minimum point is (1, 3)
(1, 3)
Intercept on the y axis
is 5
x = 1

6. Example
Think about… What is the maximum or minimum value?
Think about… What shape will the graph be?
Where is its turning point?
To sketch the graph of y x
(–2, 7)
x = –2
has a maximum value
when x = –2
Maximum point is (–2, 7) 3
Intercept on the y axis
is 3
Note you can find the intercepts on the x axis by solving

7. Completing the square 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?