These web-based activities were developed, during summer sessions from 1997 through 1999, by students at Mainland High School, located in Daytona Beach, FL. As part of Mainland's CCTT grant, the students were given permission by the book's publisher to convert the ancillary materials of the County's adopted Algebra 1 textbook (Glencoe Algebra 1 — ISBN: 9780028253282) into html format.

Modeling Mathematics
Completing the Square

This activity is found on 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. You can use algebra tiles as a model for this technique.

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

Step 1. Subtract 1 from each side of the equation. NOTE: If the coefficient of x2 had not been 1 then you would have needed to divide every term of the equation a.

x2 + 4x + 1 −1 = 0 −1

x2 + 4x = −1

then model the equation x2 + 4x = −1

Step 2. Arrange the x2-tile and the x-tiles into a square.

Step 3. In order to complete the square, you need to add 4 1-tiles to the left side of the mat. Since you are modeling an equation, you must also add 4 1-tiles to the right side of the mat.

Step 4. Remove the "zeroed pair" on the right side of the mat. You have completed the square, and the equation is x2 + 4x + 4 = 3 or (x+2)2 = 3.

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

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

x2 −6x + 5 −5 = 0 −5

x2 −6x = −5

then model the equation x2 −6x = −5

Step 2. Arrange the x2-tile and the x-tiles into a square.

Step 3. In order to complete the square, you need to add 9 1-tiles to the left side of the mat. Since you are modeling an equation, you must also add 9 1-tiles to the right side of the mat.

Why positive 1-tiles? Because (−6/2)2 = (−3)2 = +9

Step 4. Remove the "zeroed pairs" on the right side of the mat. You have completed the square, and the equation is x2 −6x + 9 = 4 or (x − 3)2 = 4.

Why (x − 3)2 when you added 9 positive 1-tiles? The − comes from the original −6x.

Summary of Steps:

  1. Make sure that the coefficient of x2, a, equals 1. If not, divide the entire equation by a.
  2. Move the constant to the other side of the equation.
  3. Divide the coefficient of x in half and square the resulting value.
  4. Add the squared value from step 3 to both sides of the equation.
  5. Factor the left side - it should be a perfect square trinomial.
  6. Solve for x by taking the square root of both sides. Don't forget to include ± in front of the constant on the right side

 

Copyright 1997-1999: html adaptation
Career Connection to Teaching with Technology