Perfect Square Trinomial — Formula & Examples

TL;DR

A perfect square trinomial is a three-term expression that factors into the square of a binomial — a² + 2ab + b² = (a + b)² or a² − 2ab + b² = (a - b)². This article shows you the formula, how to recognize one on sight, how to factor it, and six worked examples.

What Is a Perfect Square Trinomial?

A perfect square trinomial is a trinomial (a three-term polynomial) that equals the square of a binomial. In plain terms, it is what you get when you multiply a binomial like (a+b) by itself: (a + b)² = a² + 2ab + b². Two examples make the idea concrete:

1. The trinomial x² + 6x + 9 is a perfect square because it equals (x + 3)².
2. The trinomial x² − 10x + 25 is a perfect square because it equals (x − 5)².

What Is the Perfect Square Trinomial Formula?

There are two forms, separated only by the sign of the middle term. Both come directly from squaring a binomial.

Positive middle term — from squaring a sum:

(a + b)² = a² + 2ab + b².

Negative middle term — from squaring a difference:

(a - b)² = a² - 2ab + b².

Here a is the square root of the first term, b is the square root of the last term, and 2ab is the middle term.

How Do You Know If A Trinomial Is A Perfect Square?

Run three quick checks:

1. First: Is the leading term a perfect square?
2. Second: Is the constant term a perfect square?
3. Third: Does the middle term equal 2ab, twice the product of those two roots?

If all three hold, the trinomial is a perfect square.

How Do You Factor a Perfect Square Trinomial?

Factoring is the formula run in reverse. Once a trinomial passes the test above, the factored form writes itself:

1. Take the square root of the first term.
2. Take the square root of the last term.
3. Read the sign of the middle term.
4. Write the binomial squared.

Examples of Perfect Square Trinomial

Example 1

Factor x² + 8x + 16.

Final answer: (x + 4)².

Example 2

Factor x² − 12x + 36.

Final answer: (x − 6)².

Example 3

Factor 9x² − 6x + 1.

Final answer: (3x − 1)².

Example 4

Factor 16a² − 40ab + 25b².

Final answer: (4a − 5b)².

Example 5

Find the value of c that makes x² + 10x + c a perfect square trinomial.

Final answer: c = 25.

Example 6

Is 4x² + 8x + 9 a perfect square trinomial?

Final answer: No.

Key Takeaways

• A perfect square trinomial factors into a binomial squared: a² ± 2ab + b² = (a ± b)².
• Identify it by checking: first term is a square, last term is a square, and middle term equals 2ab.
• Factor by taking the roots of the first and last terms.

Practice These Before Moving On

1. Factor x² + 20x + 100.
2. Find the value of k that makes x² − 14x + k a perfect square trinomial.
3. Decide whether 25x² + 30x + 9 is a perfect square trinomial, and factor it if it is.