Factor a quadratic trinomial ax^2 + bx + c when it has rational roots, and see the discriminant and roots at the same time.
Last updated: June 2026 | By Patchworkr Team
Factoring rewrites a quadratic as a product of binomials. When the discriminant is a perfect square, the roots are rational and the trinomial factors cleanly.
If the discriminant is negative, there are no real roots. If it is not a perfect square, the roots are irrational and the trinomial does not factor over the rationals.
Example: 2x^2 + 5x + 3 = (2x + 3)(x + 1).
Factor 2x^2 + 5x + 3.
1. Compute the discriminant: 5^2 - 4 * 2 * 3 = 1.
2. Since 1 is a perfect square, the roots are rational.
3. The factors are (2x + 3)(x + 1).
Final answer: (2x + 3)(x + 1)
The calculator explains that the discriminant is not a perfect square, so the polynomial does not factor over the rationals.
The tool is designed for exact factoring of standard quadratics with integer coefficients.
It tells you how many real roots the quadratic has and whether the roots are rational or irrational.
Then the expression is not quadratic, so it is rejected as an invalid trinomial input.
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