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.

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.

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.