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Perfect Square Trinomial Calculator

Check whether ax² + bx + c is a perfect square trinomial

a =
b =
c =

Perfect Square Formula

(px+q)² = p²x² + 2pqx + q²

A perfect square trinomial is produced by squaring a binomial. The first term must be the square of px, the last term must be the square of q, and the middle term must be twice the product of those two base terms. Checking these three pieces helps determine whether a quadratic can be written in compact squared form, which is useful for factoring and completing the square.

Note: This checker is intended for numeric quadratic coefficients.

How to Recognize the Pattern

A perfect square trinomial is created by squaring a binomial. Recognizing the pattern helps factor quadratics quickly and supports completing-the-square work.

Square Ends

The first and last terms must be square terms.

Middle Term

The middle coefficient must equal 2pq or -2pq.

Positive Square

The last term q² is non-negative because it is a square.

Sign Choice

The sign inside (px±q)² follows the sign of the middle term.

💡 Example: x²-10x+25 = (x-5)².

Applications

FactoringCompleting SquareQuadratics

Frequently Asked Questions

What is a perfect square trinomial calculator?
It checks whether ax²+bx+c can be written as (px+q)² and shows the factorization steps.
What is a perfect square trinomial?
A perfect square trinomial is a quadratic such as x²+6x+9 that equals the square of a binomial.
What pattern should I look for?
The first and last terms should be perfect squares, and the middle term should equal twice their square-root product.
Is x²-10x+25 a perfect square?
Yes. It equals (x-5)² because x² and 25 are squares and -10x is 2·x·(-5).
How is this related to completing the square?
Completing the square creates a perfect square trinomial by adding and subtracting the needed square term.

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