How to Use the Difference of Two Squares Theorem to Solve Quadratic Equations

Instructor: Michael Quist

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

When a quadratic formula is made up of the difference between two perfect squares, factoring becomes much easier. In this lesson we'll explore how to use the Difference Between Two Squares Theorem to solve certain quadratic equations.

Terms Review

Algebra teachers don't always tell you, but factoring quadratic equations can be a painful experience! Fortunately for all of us non-super-math-geek types, some of those quadratics can be a LOT easier if you know a few tricks. One of the those special cases is the difference between two squares.

A quadratic equation is a special equation, usually in the form ax² + bx + c = 0, where x is an unknown, a is some number other than zero, and b and c can be any numbers. The graph of this equation is always a parabola (sort of like a soup bowl, either right-side-up or upside-down), which may or may not cross the x axis.

Factoring is the processing of breaking a quadratic equation into pieces (factors) that may be multiplied together to produce the quadratic equation. This is useful because of how the number 0 works. Anything that is multiplied by 0 becomes 0, so if you can break your quadratic into factors, you can solve for the values of x that will make it zero (the places where the graph crosses the x axis).

Difference Between Squares Theorem

The Difference of Two Squares theorem tells us that if our quadratic equation may be written as a difference between two squares, then it may be factored into two binomials, one a sum of the square roots and the other a difference of the square roots. This is sometimes shown by the expression A² - B² = (A + B) (A - B). This technique works for any expression, so long as you have a difference between squares.

One nice thing about this technique is that you can take the square root of any positive expression, so you don't have to have perfect squares for this to work. Notice, however, that this only works for a DIFFERENCE between squares, not a SUM of squares, such as x² + 4. A sum of squares may not be factored in this way.


Let's look at some examples of how to use this to solve quadratic equations.

Example 1:

Say you have the equation x² - 4 = 0. The steps are simple.

1. Write two sets of parentheses.

( ) ( )

2. In the first set, write the square root of the x² term (just x), a '+' sign, then the square root of the second term, 4. You should therefore get :

(x + 2)

3. In the second set, write the two squares again, but this time put a '-' between them.

(x - 2)

4. Either of the factors may be 0, so set up each factor as an expression equal to 0.

(x + 2) = 0; (x - 2) = 0

5. If x + 2 = 0, then subtracting 2 from both sides results in:

x = -2.

6. If x - 2 = 0, then adding 2 to both sides results in:

x = 2.

7. The variable x may be 2 or -2.

Example 2

Let's try one where a is not 1. How about 4x² - 25 = 0?

  1. 4x² - 25 = 0 (original equation)
  2. (2x + 5) (2x - 5) = 0 (sum of square roots times the difference of square roots)
  3. 2x + 5 = 0, or 2x - 5 = 0 (set each factor equal to 0)
  4. 2x = -5, or 2x = 5 (isolate the x term by transferring the non-x term to the other side)
  5. x = -5/2, or x = 5/2 (divide both sides by the coefficient of x)

Example 3

Use this same technique when the equation is a difference between two squared variables. For example, x² - y² = 0; may be factored as follows:

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