Math 101 Intermediate Algebra

Solving Quadratic Equations by Completing the Square
Chapter 9, Section 1

Square Root Property

We have been using only principal square roots, but now we'll start using both the positive and negative square root.

The square root property is that if , where a is a real number, then .

 Example: Solve  (x - 3)2 = 4. x - 3 = ±2 Square root property x = 3 ± 2 Add 3 to both sides x = 3 + 2 or x = 3 - 2 Two solutions x = 5, 1 Simplyfying

Idea of Completing the Square

Given the problem

Solve x2 + bx + c = 0,

find a number that complets the square of x2 + bx to get to an equivalent equation of the form

Solve (x - d)2 = f,

that can be solved as in the above example.

 Example: Solve  x2 - 2x - 6 = 0. x2 - 2x = 6 Rewrite equation x2 - 2x + 1 = 6 + 1 Add square of ½ coefficient of x term to both sides (this is the completing the square step) (x - 1)2 = 7 Factor left hand side into it's perfect square x - 1 = ± Square root property x = 1 ± Solve for x x = 1 + x = 1 - Two solutions

Steps to Solving a Quadratic Equation by Completing the Square

1. Make the numerical coefficient of the x2 term equal to 1.

2. Rewrite the equation with the constant by itself on the right side of the equation.

3. Take ½ the numerical coefficient of the x term, square it, and add this quantity to both sides of the equation.

4. Factor the trinomial into the square of a binomial.

5. Use the square root property to take the square root of both sides of the equation.

6. Solve for x.

7. Check the solution in the original equation.

Note that the variable was called "x" in the above steps, but it can be called by any variable name, not necessarily just x.