Completing the Square

# Completing the Square

GCSE(H),

The expression (a + 3)2 is an example of a perfect square and expands to a2 + 6a + 9.

The reverse process is called completing the square. Given b2 + 8b + 22, create a term plus a constant value that can be squared. This is useful as it gives the turning point, or vertex, of the graph of the function.

The constant term inside the bracket is half the coefficient of the b term. The square term is therefore (b + 4). Expanding (b + 4)2 gives b2 + 8b + 16. A further 6 needs to be added to this expression to obtain the original expression:

b2 + 8b + 22 = (b + 4)2 + 6.

The turning point of the quadratic graph is at (-4, 6): the value of the x-coordinate is the value of b that makes the squared term zero; the value of the y-coordinate is the value of the constant.

## Examples

1. Complete the square for 4a2 + 8a - 12.

Answer: (2a + 2)2 - 16

Take the square root of the x2 coefficient: √4 = 2. This is the new x multiplier;

For the integer; half the x coefficient: frac(1)(2) x 8 = 4; then divide by the new x multiplier: 4 ÷ 2 = 2.

Result is (2a + 2)2 = 4a2 + 8a + 4. Need to get the integer to -12, so subtract 16.

2. Complete the square for b2 - 10b.

Answer: (b - 5)2 - 25

The square root of the b2 coefficient is 1. This is the new b multiplier.

For the integer: half the b coefficient, = -5.

Therefore (b - 5)2 = b2 - 10b + 25.

Need to adjust the integer value by -25 to obtain b2 - 10b.