How to Complete the Square

Completing the square is a mathematical procedure used on quadratic equations. Quadratic equations can be written in three different forms: standard form, factored form, and vertex form.

The purpose of completing the square is to convert a quadratic equation written in standard form to vertex form.

How to Complete the Square

Completing the square of a quadratic in standard form (shown below) requires a few simple steps.

y = = ax^2 + bx + c

Step 1: Factor the Greatest Common Factor between a and b if a is not 1

y = a(x^2 + bx) + c

Step 2: Determine the Adding & Subtracting Factor

Adding & Subtracting Factor = (b/2)^2

Step 3: Add and Subtract the Factor to the Quadratic Equation

y = (ax^2) + bx + factor - factor) + c 

Step 4: Multiply the -m by a and Remove from Brackets and Subtract by c

y = (ax^2 + bx + factor) + c - factor(a)

Step 5: Factor (ax^2 + bx + factor)

y = a(x+d)^2 + e

Examples

Let's take a look at a few examples:

Example 1: x^2 + 16x + 8

Step 1: Factor the Greatest Common Factor between a and b if a is not 1

Since a = 1, we can skip this step.

Step 2:  Determine the Adding & Subtracting Factor

Adding & Subtracting Factor = (b/2)^2 = (16/2)^2 = (8)^2 = 64

Step 3: Add and Subtract the Factor to the Quadratic Equation

y= (x^2 + 16x + 8 - 8) + 8

Step 4: Multiply the -m by a and Remove from Brackets and Subtract by c

y = (x^2 + 16x + 8) + 8 - 8

y = (x^2 + 16x + 8) 

Step 5: Factor (ax^2 + bx + factor)

y = (x+8)^2

Example 2: 2x^2 + 12x + 5

Step 1: Factor the Greatest Common Factor between a and b if a is not 1

y = 2(x^2 + 6x) + 5

Step 2:  Determine the Adding & Subtracting Factor

Adding & Subtracting Factor = (b/2)^2 = (6/2)^2 = (3)^2 = 9

Step 3: Add and Subtract the Factor to the Quadratic Equation

y = 2(x^2 + 6x + 9 - 9) + 5

Step 4: Multiply the -m by a and Remove from Brackets and Subtract by c

y = 2(x^2 + 6x + 9) + 5 - 18

y = 2(x^2 + 6x + 9) - 13

Step 5: Factor (ax2 + bx + factor)

y = 2(x^2 + 6x + 9) - 13

y = 2(x+3)^2 - 13