Complete the Square

Definition: This is a technique used to solve quadratics where the portion of the equation with the independent variable is separated and expressed as the square of a binomial.

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Example: Complete the square.

x² + 4x + 9

Solution: 

Step 1: Separate the independent variable.

(x² + 4x) + 9

Step 2: Take half the coefficient of x and square it.

(4/2) = 2
2² = 4

Step 3: Complete the square by adding 4 as the new constant to the separated x-terms. Be sure to subtract it from the 9 at the end so the value of the expression remains unchanged (since + 4 – 4 = 0).

(x² + 4x + 4) + 9 – 4
(x² + 4x + 4) + 5

Step 4: Express the separated portion in the form (x + a)² where a is half the coefficient of x.

(x + 2)² + 5

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