In these lessons, we will learn how to factor perfect square trinomials.

**Related Pages**

Factoring Trinomials

Factoring Trinomials Worksheets

Algebra Lessons

Math Worksheets

The following diagrams show the factoring and expanding of Perfect Square Trinomials. Scroll down the page for examples and solutions of factoring 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 + b^{2}

Take note that:

- The first term and the last term are perfect squares.
- 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 + b^{2} = (ax + b)^{2}

The perfect square trinomial can also be in the form:

(ax)^{2} – 2abx + b^{2},

in which case it will factor as follows:

(ax)^{2} – 2abx + b^{2} = (ax <– b)^{2}

**Example:**

Factor the following trinomials:

a) x^{2} + 8x + 16

b) 4x^{2}– 20x + 25

**Solution:**

a) x^{2} + 8x + 16

= x^{2} + 2(x)(4) + 4^{2}

= (x + 4)^{2}

b) 4x^{2}– 20x + 25

= (2x)^{2}– 2(2x)(5) + 5^{2}

= (2x – 5)^{2}

**Example:**

Factor and solve the following equations:

a) x^{2} + 2x + 1 = 0

b) x^{2} + 6x + 9 = 0

**Solution:**

a)x^{2} + 2x + 1 = 0

⇒ (x + 1)^{2} = 0

⇒ x + 1 = 0

⇒ x = − 1

b) x^{2} + 6x + 9 = 0

⇒ x^{2} + 2(3)x + 3^{2} = 0

⇒ (x + 3)^{2} = 0

⇒ x + 3 = 0

⇒ x = − 3

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.

To factor a perfect square trinomial we need to be able to recognize perfect square factors.

a^{2} + 2ab + b^{2} = (a + b)(a + b) = (a + b)^{2}

a^{2} - 2ab + b^{2} = (a - b)(a - b) = (a - b)^{2}

**Example:**

Factoring

4x^{2} - 12x + 9

16x^{2} + 56x + 49

**Example:**

Factor 9x^{2} + 24xy + 16y^{2}

**Example:**

Factor 81x^{2} − 36xy + 4y^{2}

- The sign between the terms are positive or different.
- If you double the root of the first and last terms you get the middle term.

**Example:**

Determine if the given polynomial is a perfect square trinomial. If it is, factor and check your answer.

- 4x
^{2}- 20xy + y^{2} - 9x
^{2}+ 12x + 4

**Example:**

Factor (2p + t)^{2} + 6(2p + t) + 9

**Example:**

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

v^{2} + 14v + 49

t^{2} + 1/3t + 1/36

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