Completing The Square

By M Ransom

A “complete” square is a quadratic expression such as

which can be factored as the square of a term. In this case we would have

An expression such as is not a complete square because it cannot be factored as the square of a term as we had with the previous expression .

Knowing how to complete a square can be of assistance in solving equations and writing certain equations in standard form. Examples of each are shown below.

Directions and/or Common Information:

Here are three examples of using the technique of completing the square to solve equations for x.

#1 What values of x would make the following equation true

Here are the steps you should follow as you learn this factoring method.

Step 1: Square half the coefficient of the “x” term and add to both sides:

which yields

Step 2: Factor the complete square:

Step 3: Take the square root of both sides:

#2 What values of x would make the following equation true

#3 What values of x would make the following equation true

It is easier to do this if the coefficient of x² is 1. So you should first divide both sides by 2:

. Now follow the four steps outlined above in Example #1 to solve for x.

Directions and/or Common Information:

Here are four examples of using the technique of completing the square to determine the standard form of each of the classic conic sections: circles, ellipses, parabolas, and hyperbolas.

#1 Use the technique of completing the square to determine the center and radius for this circle:

Step 1: Square half the coefficient of the “x” and “y” terms and add to both sides:

Step 2: Factor the complete squares:

#2 For the ellipse

determine the co-ordinates of its center and then find the lengths of its semi-major axis, a; its semi-minor axis, b; and the distance from each focus to the center, c.

Step 1: Factor the 9 and the 4 from the terms in x and y:

Step 2: Square half the coefficient of the “x” and “y” terms within the parentheses. To add to the right side, note that you must multiply by 9 and 4 first.

Step 3: Factor the complete squares:

Step 4: Divide both sides by 36:

#3 For the hyperbola

determine the co-ordinates of its center and then find the lengths of its semi-major axis, a; semi-minor axis, b; and the distance from its center to each focus, c.