Completing The Square

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.

Example Group #1

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:

	What is your answer?

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

	What is your answer?

	#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.

	What is your answer?

Example Group #2

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:

	What is your answer?

	#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:

	What is your answer?

	#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.

	What is your answer?

	#4 For the parabola determine the co-ordinates of its vertex and whether it opens up or down. As given, this equation is not a complete square. By completing the square, this equation can be rewritten in “vertex” form as follows: Step 1: Square half the coefficient of the “x” term and add to both sides: Step 2: Factor the complete square: Step 3: Solve for y:

	What is your answer?

	#5 For the parabola determine the co-ordinates of its vertex and whether it opens up or down. It is easier to do this if the coefficient of x² is 1. So you should first factor -3 from the terms in x: Step 2: Square half the coefficient of the “x” term within the parentheses. To add to the left side, note that you must multiply by -3 first. Step 3: Factor the complete square: Step 4: Solve for y:

	What is your answer?