An alternative approach. Completing the square An alternative approach.

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.

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

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.

Example 1 (continued) Now we can substitute for p and calculate q. From the constant term p2 + q = -2 so 32 + q = -2 9 + q = -2 q = -11 giving x2 + 6x – 2 = (x + 3)2 - 11

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

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

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

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

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

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

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

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 - 1

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

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

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

Example 5 continued q + a(x+p)2 = q + ax2 + 2apx + ap2 q + ap2 + 2apx + ax2 Compare 3 + 8x - 2x2 This shows that a = -2

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