How to Factor Perfect-Square Trinomials

By Carrie Perles
An expression is only one half of an equation; expressions do not contain equal signs.
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Writing an algebraic expression as a product of two or more factors can help you to solve equations more easily. Perfect square trinomials are a special type of expression that can be factored even more easily than the trial-and-error method usually used in factoring. To understand how factoring perfect square trinomials works, you should know the following two formulas: (ax + b)^2 = a^2x^2 + 2ab + b^2 and (ax - b)^2 = a^2x^2 - 2ab + b^2. Because these are true, if you ever see a trinomial, or an expression that has three terms, that looks like it might fit these equations, you can try to factor it accordingly.

Step 1

Find the values of a and b. For example, if you are trying to factor the expression 4x^2 + 12x + 9, the value of a would be the square root of 4 and the value of b would be the square of 9, which are 2 and 3, respectively.

Step 2

Double check that the middle term in the formula above, 2ab, equals the coefficient of the middle term in the given expression. For example, (2)(2)(3) = 12, in the example above.

Step 3

Write the expression in the form (ax+b)^2 or (ax-b)^2, using the a and b values determined above. For example, if a = 2 and b = 3 the expression could be factored to (2x+3)^2.

Step 4

Check your answer by using the FOIL method (multiplying the first, outer, inner, and last terms of the two factors) to convert your answer into the original trinomial. For example, (2x+3)^2 = (2x+3)(2x+3) = 4x^2 + 6x + 6x + 9 = 4x^2 + 12x + 9, which is the original trinomial. This ensures your answer is correct.

About the Author

Keren (Carrie) Perles is a freelance writer with professional experience in publishing since 2004. Perles has written, edited and developed curriculum for educational publishers. She writes online articles about various topics, mostly about education or parenting, and has been a mother, teacher and tutor for various ages. Perles holds a Bachelor of Arts in English communications from the University of Maryland, Baltimore County.