Completing the Square

## Completing the Square

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

The reverse process - taking an expression and creating a perfect square with an additional constant - is called completing the square. This is useful as it gives the turning point of the graph of the function.

Take the expression b^2 + 8b + 22:

Halve the coefficient of the b term: 8 ÷ 2 = 4;

Create the squared term from the b and the 4: (b + 4);

Expand the squared term: (b + 4)^2 = b^2 + 8b + 16;

(b + 4)^2 + 6 = b^2 + 8b + 22

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

If the squared term has a coefficient, square root the coefficient and use this inside the bracket. Then divide the x term by that same value; and halve as before:

9x^2 + 30x + 13

√9 = 3 to get the coefficient of the x term;

and 30 ÷ 3 = 10 and halved = 5;

Create the squared term: (3x + 5);

Expand the squared term (3x + 5)^2 = 9x^2 + 30x = 25

Add a constant -12 for the original expression:

(3x + 5)^2 - 12 = 9x^2 + 30x + 13

## Example 1

Complete the square for 4a^2 + 8a - 12

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

For the integer; divide by the new x multiplier 8 / 2 = 4; then halve 4 ÷ 2 = 2

Obtain (2a + 2)^2 = 4a^2 + 8a + 4

Add a constant (-16) to obtain the original expression (2a + 2)^2 - 16 = 4a^2 +8a - 12

Answer: (2a + 2)^2 - 16

## Example 2

Complete the square for b^2 - 10b

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

Therefore (b - 5)^2 = b^2 - 10b + 25

Add a constant -25 to obtain (b - 5)^2 - 25 = b^2 - 10b

Answer: (b - 5)^2 - 25