Solving Quadratic Equations by Completing the Square
Example 1
Completing the square with a = 1
Solve x^{2} + 6x + 5 = 0 by completing the square.
Solution
The perfect square trinomial whose first two terms are x^{2} + 6x is
x^{2} + 6x + 9.
So we move 5 to the righthand side of the equation, then add 9 to each side:

= 5 
Subtract 5 from each side. 

= 5 + 9 
Add 9 to each side to get
a perfect square trinomial. 

= 4 
Factor the lefthand side. 

= Â±

Evenroot property 
x + 3 
= 2 
or 
x + 3 
= 2 

x 
= 1 
or 
x 
= 5 

Check in the original equation:
(1)^{2} + 6(1) + 5 = 0 and
(5)2 + 6(5) + 5 = 0
The solution set is {1, 5}.
Caution
The perfect square trinomial that we have used so far
had a leading coefficient of 1. If a ≠ 1, then we must divide each side of the equation
by a to get an equation with a leading coefficient of 1.
The strategy for solving a quadratic equation by completing the square is stated
below.
Strategy for Solving Quadratic Equations
by Completing the Square
1. The coefficient of x^{2} must be 1.
2. Get only the x^{2} and the x terms on the lefthand side.
3. Add to each side the square of
the coefficient of x.
4. Factor the lefthand side as the square of a binomial.
5. Apply the evenroot property.
6. Solve for x.
7. Simplify.
In our procedure for completing the square the coefficient of x^{2} must be 1.
We can solve ax^{2} + bx + c = 0 with a ≠ 1 by completing the square if we first
divide each side of the equation by a.
