Perfect Square Trinomial Calculator

A perfect square trinomial is a quadratic expression of the form ax² + bx + c that can be written as the square of a binomial — i.e., (px + q)². This happens when the discriminant b² − 4ac equals zero, meaning the expression has a repeated root. See also our Absolute Value Equation.

Check three conditions: (1) both a and c must be perfect squares (or their square roots must be rational), (2) a and c must have the same sign, and (3) b must equal ±2·√a·√c. Equivalently, the discriminant b² − 4ac must equal exactly zero.

Once you confirm the trinomial is a perfect square, find p = √a and q = √c. The factored form is (px + q)² if b is positive, or (px − q)² if b is negative. For example, 4x² − 12x + 9 factors as (2x − 3)².

Yes. Here a = 1, b = 4, c = 4. The discriminant is 4² − 4(1)(4) = 16 − 16 = 0, confirming it is a perfect square. It factors as (x + 2)².

No. The discriminant is 2² − 4(1)(2) = 4 − 8 = −4, which is not zero. A non-zero discriminant means the trinomial cannot be expressed as the square of a binomial with rational coefficients.

Start with any value of a and c (both positive or both negative), then set b = ±2·√|a|·√|c|. This technique — called completing the square — is widely used to solve quadratic equations and convert parabolas to vertex form.

The general formula is (px ± q)² = p²x² ± 2pqx + q², where a = p², b = ±2pq, and c = q². The signs of a and c are always the same, and |b| = 2·√a·√c.

The coefficient a cannot be negative in the standard real-number sense, since a = p² ≥ 0. However, you may encounter expressions like −(x + 3)² = −x² − 6x − 9, where all three coefficients are negative. This calculator checks for real-valued perfect square trinomials where a > 0.