Perfect Square Trinomial Calculator

Enter the coefficients of ax² + bx + c to check whether it is a perfect square trinomial and get its factored form.

Quick Facts

Sum form
(m + n)² = m² + 2mn + n²
Positive middle term when the binomial is a sum.
Difference form
(m - n)² = m² - 2mn + n²
Negative middle term when the binomial is a difference.
Perfect-square test
b² = 4ac
Holds exactly when a ≥ 0, c ≥ 0, and the discriminant is zero.
Completing the square
x² + bx + (b/2)² = (x + b/2)²
Turns any x² + bx expression into a perfect square trinomial.

Your Results

Calculated
Perfect Square Trinomial?
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Passes the b² = 4ac test
Factored Form
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(mx ± n)²
Discriminant (b² - 4ac)
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Zero means a repeated root
Repeated Root (x)
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x = -b / 2a, when applicable

Ready

Enter a, b, and c, then press Calculate.

Formula and Method for Perfect Square Trinomials

A perfect square trinomial is a three-term polynomial that comes from squaring a binomial. Squaring (mx + n) gives (mx + n)² = m²x² + 2mnx + n², and squaring (mx - n) gives (mx - n)² = m²x² - 2mnx + n². So a general trinomial ax² + bx + c is a perfect square trinomial exactly when a and c are non-negative (so they have real square roots m = √a and n = √c) and the middle term satisfies b = 2mn or b = -2mn. This calculator applies that test to your a, b, c values and returns the factored form when it holds.

The three-term test

To check any trinomial ax² + bx + c by hand: (1) take the square root of the first term to get m = √a; (2) take the square root of the last term to get n = √c; (3) double the product 2mn and compare it to the middle term b. If b matches 2mn or -2mn, the trinomial factors as (mx + n)² or (mx - n)² respectively. This is equivalent to checking that the discriminant b² - 4ac equals zero, since a quadratic with a zero discriminant has one repeated real root and its graph is tangent to the x-axis at that point.

Completing the square

Perfect square trinomials are the foundation of "completing the square": starting from x² + bx, adding (b/2)² produces the perfect square trinomial x² + bx + (b/2)² = (x + b/2)². This is exactly how the quadratic formula is derived, and it's the standard way to rewrite a quadratic in vertex form or to solve equations that don't factor with simple integers.

Common mistakes

  • Forgetting the sign check: a and c being perfect squares is not enough — the middle term must equal exactly 2mn or -2mn, not just "close" to it.
  • Ignoring a negative leading coefficient: if a is negative, there is no real m with m² = a, so the trinomial cannot be written as (mx + n)² even if b² = 4ac happens to hold algebraically.
  • Mixing up the discriminant sign convention: the test is b² - 4ac = 0, not b² + 4ac; a positive or negative discriminant means the trinomial is not a perfect square.

Real-world applications

  • Factoring quadratics quickly in algebra courses, without needing the full quadratic formula.
  • Completing the square to convert a quadratic to vertex form for graphing parabolas.
  • Deriving and understanding the quadratic formula itself, which is built by completing the square on ax² + bx + c = 0.
  • Simplifying expressions in calculus and physics where squared binomials appear, such as in distance and energy formulas.

Frequently Asked Questions

What is a perfect square trinomial?
A perfect square trinomial is a three-term polynomial of the form m²x² + 2mnx + n² that results from squaring a binomial: (mx + n)² = m²x² + 2mnx + n², or (mx - n)² = m²x² - 2mnx + n². In the general trinomial ax² + bx + c, that means a and c must be perfect squares and b must equal ±2√(ac).
How do you know if a trinomial is a perfect square?
Take m = √a and n = √c (both must be real, so a and c must be zero or positive), then check whether b equals 2mn or -2mn. Equivalently, ax² + bx + c is a perfect square trinomial exactly when its discriminant b² - 4ac equals 0 and a is positive, since a zero discriminant means the quadratic has one repeated real root.
What is the difference between (a + b)² and (a - b)²?
(a + b)² = a² + 2ab + b² has a positive middle term, while (a - b)² = a² - 2ab + b² has a negative middle term; both expand to the same first and last terms. The sign of the middle term in a perfect square trinomial tells you whether the original binomial was a sum or a difference.
How does completing the square relate to perfect square trinomials?
Completing the square turns any quadratic expression x² + bx into a perfect square trinomial by adding (b/2)²: x² + bx + (b/2)² = (x + b/2)². This technique is used to derive the quadratic formula, convert a quadratic to vertex form, and solve equations that do not factor easily.

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