Factoring Perfect Square Trinomials

In this lesson, we will learn how to factor perfect square trinomials.

Related Topics:
Factoring Other Types of Trinomials

Worksheets for factoring Perfect Square Trinomials

Perfect Square Trinomials

In some cases recognizing some common patterns in the trinomial will help you to factor it faster. For example, we could check whether the trinomial is a perfect square.

A perfect square trinomial is of the form:

(ax)2 + 2abx + b2

Take note that

1. The first term and the last term are perfect squares
2. The coefficient of the middle term is twice the square root of the last term multiplied by the square root of the coefficient of the first term.

When we factor a perfect square trinomial, we will get

(ax)2 + 2abx + b2 = (ax + b)2

The perfect square trinomial can also be in the form:

(ax)2 2abx + b2

In which case it will factor as follows:

(ax)2 2abx + b2 = (ax b)2

Example 1: x2 + 2x + 1 = 0
  (x + 1)2 = 0
  x= -1
Example 2:

x2 + 6x + 9 = 0

  x2 + 2(3)x + 32 = 0
  (x + 3)2 = 0
  x = -1


Factor the following trinomials:

a) x2 + 8x + 16
b) 4x2– 20x + 25


a) x2 + 8x + 16
= x2 + 2(x)(4) + 42
= (x + 4)2

b) 4x2– 20x + 25
= (2x)2– 2(2x)(5) + 52
= (2x – 5)2



Perfect Square Trinomials
One special case when trying to factor polynomials is a perfect square trinomial. Unlike a difference of perfect squares, perfect square trinomials are the result of squaring a binomial. It's important to recognize the form of perfect square trinomials so that we can easily factor them without going through the steps of factoring trinomials, which can be very time consuming.

Factor 9x2 + 24xy +16y2

Factor 81x2 − 36xy + 4y2


Factor (2p + t)2 + 6(2p + t) + 9


This video shows how to factor the following perfect square trinomials.

v2 + 14v + 49
t2 + 1/3t + 1/36


You can use the Mathway widget below to practice Algebra or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.

(Clicking "View Steps" on the answer screen will take you to the Mathway site, where you can register for a free ten-day trial of the software.)

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