How do I know if a trinomial is a perfect square?

Check three things: the first term is a perfect square, the last term is a perfect square, and the middle term equals 2ab where a and b are the square roots of the first and last terms. If all three conditions hold, it's a perfect square trinomial.

What is the difference between a^2 + 2ab + b^2 and a^2 + 2ab - b^2?

The first is a perfect square trinomial that factors as (a+b)^2. The second is not a perfect square trinomial because the last term is negative. Perfect square trinomials must have both a^2 and b^2 as positive perfect squares; they cannot have a negative constant term.

Can I use perfect square trinomials if the leading coefficient is not 1?

Yes! As long as the first term is a perfect square (like 4x^2 = (2x)^2 or 9x^2 = (3x)^2), you can still use the pattern. Extract the square root of the leading coefficient and the constant, check that 2ab matches the middle term, and factor. For example, 4x^2 - 12x + 9 = (2x - 3)^2.

Why is it called a perfect square trinomial?

Because the trinomial factors into a perfect square—a binomial multiplied by itself, like (a+b)^2 or (a-b)^2. The word 'perfect' emphasizes that it is an exact square, not an approximation or remainder.

Perfect Square Trinomial

Quickly factor trinomials of the form $a^2 \pm 2ab + b^2$ by recognizing the perfect square pattern that gives $(a \pm b)^2$ .

a^2 \pm 2ab + b^2 = (a \pm b)^2

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What each symbol means

What Perfect Square Trinomial takes

Symbol	Meaning
$a$	The value or expression that, when squared, produces the first term of the trinomial; misidentifying $a$ causes the entire factorization to fail.
$b$	The value or expression that, when squared, produces the last term of the trinomial; confusing $b$ with $a$ or making an arithmetic error here ruins the middle coefficient $2ab$ .

When to use it

When you see a trinomial with two perfect square terms and need to factor it quickly.

Level

Usually taught in: Algebra I

Worked examples

1. Factor a trinomial with small integer coefficients

Problem

Factor

x^2 + 6x + 9

$a = x, \quad b = 3 \quad (\text{since } a^2 = x^2 \text{ and } b^2 = 9)$
We identify $a$ from the first term $x^2$ and $b$ from the last term $9$ .
$2ab = 2 \cdot x \cdot 3 = 6x$
The middle term must equal $2ab = 6x$ , which it does, confirming this is a perfect square trinomial.
$x^2 + 6x + 9 = (x + 3)^2$
Since the pattern matches, we write the factorization directly as $(a + b)^2$ with $a = x$ and $b = 3$ .

Answer: $(x + 3)^2$

This is a textbook perfect square trinomial—the first and last terms are perfect squares, and the middle term equals exactly $2ab$ . We recognized the pattern and factored directly.

2. Factor a trinomial with a leading coefficient greater than 1

Problem

Factor

4x^2 - 12x + 9

$a = 2x, \quad b = 3 \quad (\text{since } a^2 = (2x)^2 = 4x^2 \text{ and } b^2 = 9)$
The first term is $4x^2 = (2x)^2$ , so $a = 2x$ ; the last term is $9 = 3^2$ , so $b = 3$ .
$2ab = 2 \cdot 2x \cdot 3 = 12x$
With $a = 2x$ and $b = 3$ , the product $2ab = 12x$ . Our trinomial has $-12x$ , so we use the minus version of the pattern.
$4x^2 - 12x + 9 = (2x - 3)^2$
The trinomial matches $a^2 - 2ab + b^2 = (a - b)^2$ with $a = 2x$ and $b = 3$ .

Answer: $(2x - 3)^2$

Even though the leading coefficient is not 1, we can still recognize the perfect square pattern by taking square roots of the first and last terms to find $a$ and $b$ . The minus sign in the middle tells us to use $(a - b)^2$ instead of $(a + b)^2$ .

3. Factor a trinomial in a word problem about basketball scoring

Problem

A basketball player's scoring follows a pattern: in week

n

, she scores

n^2 + 4n + 4

points total. Factor this expression to find another way to describe her scoring.

$a = n, \quad b = 2 \quad (\text{since } a^2 = n^2 \text{ and } b^2 = 4)$
The scoring formula $n^2 + 4n + 4$ has first term $n^2$ (so $a = n$ ) and last term $4$ (so $b = 2$ ).
$2ab = 2 \cdot n \cdot 2 = 4n$
We verify that the middle term matches: $2ab = 4n$ , which is exactly what we have in the trinomial.
$n^2 + 4n + 4 = (n + 2)^2$
The trinomial matches $a^2 + 2ab + b^2 = (a + b)^2$ , so her scoring in week $n$ equals $(n + 2)^2$ .

Answer: $(n + 2)^2$

By recognizing this trinomial as a perfect square, we see that her weekly scoring in week $n$ equals $(n + 2)^2$ . For instance, in week 3 she scores $(3+2)^2 = 25$ points, and in week 5 she scores $(5+2)^2 = 49$ points. The perfect square form makes predictions about her scoring pattern much easier.

Common mistakes

Where Perfect Square Trinomial usually goes wrong

The mistake	Why it is wrong	The fix
Factoring $x^2 + 5x + 6$ as $(x + 2.5)^2$ because the first term is a perfect square.	Not every trinomial with a perfect square first term is a perfect square trinomial; the last term must also be a perfect square, and the middle term must equal exactly $2ab$ .	Check: is the last term a perfect square? Here $6$ is not a perfect square. Use Factoring Trinomials instead: $(x + 2)(x + 3)$ .
Factoring $x^2 + 7x + 9$ as $(x + 3)^2$ even though the middle term should be $2(1)(3) = 6x$ , not $7x$ .	The pattern $a^2 + 2ab + b^2$ is rigid; if the middle coefficient does not equal exactly $2ab$ , the trinomial is not a perfect square, and this factorization is incorrect.	Check: does $2ab = 2(x)(3) = 6x$ equal the middle term $7x$ ? No. Use Factoring Trinomials or the Quadratic Formula instead.
Writing $4x^2 - 12x + 9 = (4x - 9)^2$ by taking $a = 4x$ and $b = 9$ directly from the terms.	The bases $a$ and $b$ are the square roots of the first and last terms, not the terms themselves. Here $a = \sqrt{4x^2} = 2x$ and $b = \sqrt{9} = 3$ , not $4x$ and $9$ .	Take square roots: $\sqrt{4x^2} = 2x$ and $\sqrt{9} = 3$ . Check: $2ab = 2(2x)(3) = 12x$ , which matches the middle term, confirming $(2x - 3)^2$ .

Tips and when to use something else

Always verify that all three pieces fit the pattern: the first term must be a perfect square, the last term must be a perfect square, and the middle term must equal exactly $2ab$ .
Perfect square trinomials always have both a $+$ and $-$ version: $a^2 + 2ab + b^2 = (a+b)^2$ and $a^2 - 2ab + b^2 = (a-b)^2$ ; the sign of the middle term determines which form to use.
If the trinomial does not fit the pattern, switch to Factoring Trinomials (using the AC method or trial-and-error) or the Quadratic Formula instead of forcing a perfect square factorization.
Mastering perfect square trinomials speeds up algebra and prepares you for completing the square in quadratics and simplifying radicals in calculus.

Related formulas

Factoring TrinomialsBreak a trinomial into two binomials by finding two numbers that add to b and multiply to c; use when solving equations or simplifying.Difference of SquaresInstantly factor binomials that are differences of perfect squares by recognizing the pattern and splitting into a product of a sum and difference.Distributive PropertyThe Distributive Property lets you multiply a number by a sum or difference by distributing the multiplication to each term separately.

Frequently asked questions

How do I know if a trinomial is a perfect square?: Check three things: the first term is a perfect square, the last term is a perfect square, and the middle term equals $2ab$ where $a$ and $b$ are the square roots of the first and last terms. If all three conditions hold, it's a perfect square trinomial.
What is the difference between $a^2 + 2ab + b^2$ and $a^2 + 2ab - b^2$ ?: The first is a perfect square trinomial that factors as $(a+b)^2$ . The second is not a perfect square trinomial because the last term is negative. Perfect square trinomials must have both $a^2$ and $b^2$ as positive perfect squares; they cannot have a negative constant term.
Can I use perfect square trinomials if the leading coefficient is not 1?: Yes! As long as the first term is a perfect square (like $4x^2 = (2x)^2$ or $9x^2 = (3x)^2$ ), you can still use the pattern. Extract the square root of the leading coefficient and the constant, check that $2ab$ matches the middle term, and factor. For example, $4x^2 - 12x + 9 = (2x - 3)^2$ .
Why is it called a perfect square trinomial?: Because the trinomial factors into a perfect square—a binomial multiplied by itself, like $(a+b)^2$ or $(a-b)^2$ . The word 'perfect' emphasizes that it is an exact square, not an approximation or remainder.

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Reviewed 2026-09-18