# Perfect Square Trinomial Calculator

URL: https://speedycalculator.com/calculators/math/perfect-square-trinomial/
Last updated: 2026-10-30

> The perfect square trinomial calculator checks whether ax² + bx + c is the square of a binomial and, if so, writes it as (√a·x ± √c)².

## Inputs

- a
- b
- c

## Results

- Is it a perfect square?
- As a squared binomial
- Reason

## Formula

```
ax² + bx + c is a perfect square ⇔ b² − 4ac = 0 (with a, c ≥ 0)
(px + q)² = p²x² + 2pqx + q²
```

## Explanation

Perfect square trinomials such as x² + 6x + 9 = (x + 3)² are the building blocks of completing the square and appear whenever a quadratic has a double root.

### How to use the Perfect Square Trinomial Calculator

- **a**
- **b**
- **c**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

The discriminant of a perfect square is zero. The calculator tests that condition, then takes square roots of the first and last coefficients and uses the sign of b for the middle sign.

## Worked example

Inputs — a: 4; b: 12; c: 9.

Results — **Is it a perfect square? = Yes – it is a perfect square trinomial**; As a squared binomial = (2x + 3)²; Reason = b² − 4ac = 0 = 0, and a, c are non-negative, so a·x² + b·x + c = (√a·x ± √c)²..

## Key terms

- **Perfect square trinomial**: A trinomial equal to the square of a binomial.

## FAQ

**How do I recognise a perfect square trinomial?**
The first and last terms are squares and the middle term is twice the product of their square roots.

**Is x² + 4x + 4 a perfect square?**
Yes: (x + 2)².

**Is x² + 4x + 8 a perfect square?**
No – the discriminant 16 − 32 is not zero.

**What is the double root?**
x = −b/(2a); the parabola touches the x-axis there.


## Sources

- [OpenStax – College Algebra 2e](https://openstax.org/details/books/college-algebra-2e)
- [OpenStax – Intermediate Algebra 2e](https://openstax.org/details/books/intermediate-algebra-2e)