Factoring Trinomials Calculator
Enter the coefficients a, b, and c of the trinomial ax² + bx + c. The calculator finds the roots using the quadratic formula and rewrites the expression in factored form a(x − r₁)(x − r₂), with a full step-by-step solution and a graph of the parabola.
(x − 3)(x − 2)
Write the trinomial
Calculate the discriminant
Apply the quadratic formula
Roots
Factored form
- 1
b²
(-5)² = -5 × -5 = 25 - 2
4ac
4 × 1 × 6 = 24 - 3
Discriminant D = b² − 4ac
25 − (24) = 1
How does this calculator work?
To factor ax² + bx + c: compute discriminant D = b² − 4ac. If D ≥ 0, the roots are x = (−b ± √D)/(2a) and the factored form is a(x − r₁)(x − r₂) (or a(x − r)² for D = 0). If D < 0, no real factors exist. Enter coefficients a, b, c for the factored form, roots, and a parabola graph.
Formula
How this is calculated
A quadratic trinomial ax² + bx + c is factored by finding its roots — the x-values where the parabola crosses the x-axis. The standard method uses the quadratic formula: x = (−b ± √(b² − 4ac)) / (2a). The expression under the square root, D = b² − 4ac, is the discriminant. If D > 0 the trinomial has two distinct real roots and factors into a(x − r₁)(x − r₂). If D = 0 there is a single repeated root and the factor is a(x − r)². If D < 0 the roots are complex and the trinomial has no real factors.
Once the roots r₁ and r₂ are found the factored form follows directly from the factor theorem: any polynomial has (x − r) as a factor whenever r is a root. Multiplying a(x − r₁)(x − r₂) back out recovers the original ax² + bx + c. The calculator also tries to express roots as exact fractions when the denominator is small (≤ 20), giving cleaner results for integer-coefficient trinomials.
The graphed parabola confirms the factoring visually: its x-intercepts are the roots, its vertex lies at x = −b/(2a), and it opens upward if a > 0 or downward if a < 0.
Frequently asked questions
A negative discriminant (D < 0) means the parabola does not intersect the x-axis. The trinomial has no real roots and cannot be factored using real numbers. It can be factored over the complex numbers as a(x − z₁)(x − z₂) where z₁, z₂ are complex conjugates, but that is outside the scope of real-number algebra.
Yes — for integer-coefficient trinomials where a = 1 you can look for two numbers that multiply to c and add to b. This is faster by hand for simple cases. The quadratic formula works for all cases including non-integer and irrational coefficients, which is why this calculator uses it as the general method.
If a = 0 the expression reduces to bx + c — a linear equation, not a trinomial. Division by a is required in the quadratic formula, so a must be non-zero. Enter any non-zero value for a.
Also known as
TG we-Calculate Editorial Team. (2026). Factoring Trinomials Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/factoring-trinomials-calculator
TG we-Calculate Editorial Team. "Factoring Trinomials Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/factoring-trinomials-calculator.
TG we-Calculate Editorial Team, "Factoring Trinomials Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/factoring-trinomials-calculator
@misc{wecalculate_factoring_trinomials_calculator, title = {Factoring Trinomials Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/factoring-trinomials-calculator}}, year = {2026}, note = {TG we-Calculate} }
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