Perfect Square Trinomial Calculator
Use our perfect square trinomial calculator to calculate perfect square trinomial forms and factor them fast. Learn what is a perfect square trinomial, what makes a trinomial a perfect square, how to find perfect square trinomial patterns, the perfect square trinomial formula, and see perfect square trinomial formula examples.
What is a Perfect Square Trinomial?
A perfect square trinomial is a trinomial that can be written as the square of a binomial. In other words, it factors into something like (a + b)² or (a − b)².
If you’re asking what makes a trinomial a perfect square, it follows a very specific pattern: the first and last terms are perfect squares, and the middle term is twice the product of their square roots (with a matching sign).
A perfect square trinomial calculator helps you recognize the pattern and factor quickly, which is why it’s often used as a factor perfect square trinomial calculator in algebra.
Perfect Square Trinomial Formula
Perfect square trinomials come from expanding (a ± b)². These identities are the core formula for perfect square trinomial problems.
Middle term is +2ab.
Middle term is -2ab.
Because 25 = 5^2 and 10x = 2·x·5.
Because 4x^2 = (2x)^2, 9 = 3^2, and -12x = -2·(2x)·3.
How to Use the Perfect Square Trinomial Calculator
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Enter the trinomial expression (for example, x^2 + 10x + 25).
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The calculator checks if the first and last terms are perfect squares.
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It verifies whether the middle term matches ±2ab.
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If it matches, it returns the factored form (a ± b)^2.
Frequently Asked Questions
It’s a trinomial that factors into a binomial squared, like (a + b)^2 or (a − b)^2.
x^2 + 6x + 9 is a perfect square trinomial because it factors as (x + 3)^2.
Check whether the first and last terms are perfect squares, then see if the middle term equals ±2 times the product of their square roots.
It must match the pattern a^2 ± 2ab + b^2. The middle term must be exactly twice the product of a and b (with the correct sign).
Take square roots of the first and last terms to get a and b, then write (a ± b)^2, using the sign from the middle term.
A perfect square trinomial equals (a ± b)^2, which is always ≥ 0 for real numbers. Individual terms or the middle term can be negative (like a^2 − 2ab + b^2), but the expression as a whole is not negative for real inputs.
Those that match a^2 ± 2ab + b^2. If the middle term doesn’t match ±2ab exactly, it’s not a perfect square trinomial.
Yes. It identifies the pattern and returns the factored binomial-squared form when the trinomial is a perfect square.