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