Factor The Following Perfect Square Trinomial Calculator

Factoring perfect square trinomials is a fundamental algebra skill. This calculator helps you factor expressions like x² + 6x + 9 quickly and accurately. Learn the formula, step-by-step process, and common pitfalls to master this algebraic technique.

What is a perfect square trinomial?

A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. It has the general form:

General Form

(a + b)² = a² + 2ab + b²

(a - b)² = a² - 2ab + b²

Where:

a² is the first term
b² is the last term
2ab is the middle term (with a positive sign for addition, negative for subtraction)

Examples of perfect square trinomials include x² + 10x + 25 and y² - 12y + 36.

How to factor perfect square trinomials

Follow these steps to factor a perfect square trinomial:

Identify the first and last terms (a² and b²)
Find the square roots of these terms to get a and b
Determine the sign of the middle term (positive or negative)
Write the binomial squared with the correct sign

Example

Factor x² + 8x + 16:

First term: x² → a = x
Last term: 16 → b² = 16 → b = 4
Middle term: +8x → positive sign
Factored form: (x + 4)²

Tip

Always check your work by expanding the binomial to ensure you get back to the original trinomial.

Examples of factoring perfect square trinomials

Here are three worked examples:

Example 1

Factor: 9x² + 12x + 4

First term: 9x² → a = 3x
Last term: 4 → b² = 4 → b = 2
Middle term: +12x → positive sign
Factored form: (3x + 2)²

Example 2

Factor: y² - 10y + 25

First term: y² → a = y
Last term: 25 → b² = 25 → b = 5
Middle term: -10y → negative sign
Factored form: (y - 5)²

Example 3

Factor: 16z² - 24z + 9

First term: 16z² → a = 4z
Last term: 9 → b² = 9 → b = 3
Middle term: -24z → negative sign
Factored form: (4z - 3)²

Common mistakes to avoid

When factoring perfect square trinomials, watch out for these common errors:

Incorrectly identifying a and b - always take the square root of the first and last terms
Miscounting the sign of the middle term - it should match the sign between a and b in the binomial
Forgetting to square the binomial - the factored form must be squared
Not checking your work by expanding the binomial

Remember

The middle term must be exactly twice the product of a and b to be a perfect square trinomial.

FAQ

What is the difference between a perfect square trinomial and a perfect square binomial?: A perfect square binomial is already squared, like (x + 3)². A perfect square trinomial is the expanded form, like x² + 6x + 9.
How do I know if a trinomial is a perfect square?: Check if the first and last terms are perfect squares and the middle term is twice the product of their square roots with the correct sign.
Can all quadratic trinomials be factored as perfect squares?: No, only those that meet the perfect square trinomial pattern can be factored this way. Others may factor into two different binomials.
What if the first term is not a perfect square?: You can still factor it as a perfect square trinomial if the entire expression can be written as a square of a binomial.
How do I factor a perfect square trinomial with fractions?: Follow the same steps, but be careful with the signs and the middle term which should be twice the product of the square roots.