Lesson 5: Completing the Square

Property

To complete the square of $x^2 + bx$:

Step 1. Identify $b$, the coefficient of $x$.

Step 2. Find $(\frac{1}{2}b)^2$, the number to complete the square.

Property

The process of completing the square works best when the leading coefficient is one.
If the $x^2$ term has a coefficient $a$ other than 1, you must first take a preliminary step to make the coefficient equal to one.

To solve an equation of the form $ax^2 + bx + c = 0$:
Step 1. Divide both sides of the equation by the leading coefficient, $a$. This gives an equation of the form $x^2 + (b/a)x + (c/a) = 0$.
Step 2. Proceed with the standard steps for completing the square.

Examples

• To solve $2x^2 + 8x - 10 = 0$, first divide by 2 to get $x^2 + 4x - 5 = 0$. Then solve by completing the square: $x^2 + 4x = 5 \implies (x+2)^2 = 9 \implies x+2 = \pm 3$. The solutions are $x=1$ and $x=-5$.