1. An alternative approach.
Completing the square
An alternative approach.

2. To complete the square we are required to write ax2 + bx +c in the form
a(x + p)2 + q.
We will do this by expanding the second expression and comparing coefficients.

3. a = 1 Example 1 Write x2 + 6x – 2 in the form (x + p)2 + q
Start by expanding the bracket.
(x + p)2 + q =

4. Example 1 (continued) x2 + 2px + p2 + q Compare x2 + 6x - 2
From the coefficient of x we can see that
2p = 6 and so p = 3.

5. Example 1 (continued) Now we can substitute for p and calculate q.
From the constant term
p2 + q = -2
so q = -2
9 + q = -2
q = -11
giving x2 + 6x – 2 = (x + 3)2 - 11

6. Example 2 Write x2 – 3x + 1 in the form (x + p)2 + q.
Again start by expanding the bracket:
(x + p)2 + q =

7. Example 2 (continued) x2 + 2px + p2 + q Compare x2 – 3x + 1
From the coefficient of x we see that
2p = -3 and so p = -3/2
Substituting for p to calculate q gives (-3/2)2 + q = 1

8. Example 2 (continued)
So that x2 – 3x + 1 =

9. a > 1 or a < 1
The important difference this time is in the expansion of
a(x + p)2 + q.
We get an extra factor of a in the first three terms;
ax2 + 2apx + ap2 + q

10. Example 3 Express 3x2 + 12x – 2 in the form a(x + p)2 + q.
Start by expanding the bracket:
a(x2 + 2px + p2) + q which becomes

11. Example 3 (continued) ax2 + 2apx + ap2 + q Compare 3x2 + 12x - 2
We see that a = 3
2ap = 12
6p = 12
giving p = 2

12. Example 3 (continued) Finally from the constant term ap2 + q = -2
3 x 22 + q = -2
12 + q = -2
q = -14
3x2 + 12x – 2 = 3(x+2)2 - 14

13. Example 4 Express 4x2 + 12x – 1 in the form a(x + p)2 + q.
Expand the bracket:
a(x + p)2 = ax2 + 2apx + ap2 + q
Compare 4x2 + 12x

14. Example 4 (continued) We see that a = 4
From the middle term we see that 2ap = 12
8p = 12
p = 1.5 or 3/2
Finally ap2 + q = -1

15. Example 4 (continued) 9 + q = -1 q = -10
So 4x2 +12x – 1 = 4(x+3/2)2 - 10

16. Example 5 Complete the square for 3 + 8x – 2x2
We can swap the order around this time, and make it
q + a(x + p)2

17. Example 5 continued q + a(x+p)2 = q + ax2 + 2apx + ap2
q + ap2 + 2apx + ax2
Compare x x2
This shows that a = -2

18. Example 5 continued 2ap = 8 so -4p = 8 and p = -2
q + ap2 = q + (-2) x (-2)2
= q – 8
q = 11
3 + 8x – 2x2 = 11 – 2 (x - 2)2