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Factoring A Difference Between Two Squares

Take a look at the problem (expression) below:

4x2 - 16

The first step at factoring this is to make sure that the expression is a difference between squares.

Question Answer and Reason
Are there only two terms? Yes. The first term is 4x2; the second term is -16.
Are both coefficients (4 and 16) perfect squares? Yes. Notice 2 times 2 equals 4, and 4 times 4 equals 16.
Are all of the variables in the expression raised to an even (2,4,6, ...) power? Yes. There is only one variable, x, and it has a power of 2 which is even.
Does one term have a positive coefficient, and another term have a negative coefficient? Yes. The coefficient 4 is positive, and the coefficient -16 is negative.

Because "Yes" was answered to each of the above questions, we know that the expression is a difference between two squares. Begin the factoring process by writing two sets of open parentheses:

(                )(                )

Now find the square root of 4x2, the first term, by finding the square root of 4 and then dividing each exponent by 2. The square root of 4 is 2. Half of the exponent 2 is 1, thus x2 becomes x1 or x. Thus, the square root of the entire term is 2x. Write this term on the left inside of each set of parentheses.

(2x           )(2x           )

We will now consider 16, the second term without the negative sign. We will apply the same process that we applied to 4x2. There are no variables in 16, so we simply find that the square root of 16 is 4. Now 4 is written on the right inside of each set of parentheses.

(2x         4)(2x         4)

Add a plus sign to the middle of the first set of parentheses, then add a minus sign to the middle of the second set of parentheses.

(2x + 4)(2x - 4)

The result is two parentheses which can be multiplied to get the original expression 4x2 - 16. To check that this answer is correct, you can apply the FOIL Method which was presented in an earlier lesson.

Proceed to the next page to see another example of factoring a Difference Between Two Squares.


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