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: $(b + 4)^2 = b^2 + 8b + 16$ ;

Add an adjustment (6) to obtain the original expression:

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

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: $(b - 5)^2 - 25$