Modeling Mathematics

Completing the Square

This activity corresponds to page 742 in your textbook.

One way to solve a quadratic equation is by completing the square. To use this method, the quadratic expression on one side of the equation must be a perfect square trinomial. You can use algebra tiles as a model for this technique.

Example 1: Use algebra tiles to complete the square for x2 + 4x + 1 = 0

Step 1: Subtract 1 from each side of the equation.

x2 + 4x + 1 − 1 = 0 − 1
x2 + 4x = −1

Model the equation x2 + 4x = −1 using tiles.

Step 1 Tile Model
Step 2: Arrange the x2-tile and x-tiles into a square.
Step 2 Tile Model
Step 3: In order to complete the square, add 4 positive 1-tiles to the left side of the mat. Add 4 positive 1-tiles to the right side to maintain equality.
Step 3 Tile Model
Step 4: Remove the “zero pair” on the right side. You have completed the square:

x2 + 4x + 4 = 3
(x + 2)2 = 3

Step 4 Tile Model

Example 2: Use algebra tiles to complete the square for x2 − 6x + 5 = 0

Step 1: Subtract 5 from each side of the equation.

x2 − 6x + 5 − 5 = 0 − 5
x2 − 6x = −5

Example 2 Step 1
Step 2: Arrange the x2-tile and negative x-tiles into a square.
Example 2 Step 2
Step 3: Add 9 positive 1-tiles to both sides of the mat.
Why +9? Because (−6/2)2 = (−3)2 = +9.
Example 2 Step 3
Step 4: Remove the zero pairs. The completed square is:

x2 − 6x + 9 = 4
(x − 3)2 = 4

Example 2 Step 4

Summary of Steps for Completing the Square

  1. Make sure that the coefficient of x2, a, equals 1. If not, divide the entire equation by a.
  2. Move the constant term to the other side of the equation.
  3. Divide the coefficient of x by 2 and square the result: (b/2)2.
  4. Add the squared value from Step 3 to both sides of the equation.
  5. Factor the left side — it is now a perfect square trinomial (x + b/2)2.
  6. Solve for x by taking the square root of both sides. Do not forget to include ± in front of the constant on the right side.