Quadratic Factoring Calculator
Quadratic Factoring Calculator
Enter the coefficients a, b, and c from your quadratic expression. This finds the roots using the quadratic formula and writes the fully factored form a(x − r₁)(x − r₂) — or tells you clearly when the expression can't be factored over the real numbers.
The roots r₁ and r₂ are found using the quadratic formula and then plugged directly into the factored form, which always expands back to exactly the original expression — a useful way to check factoring work by hand. Unlike textbook examples designed to factor into neat whole numbers, real-world quadratics often have irrational or complex roots that don't factor "nicely" at all, and this calculator reports those cases plainly rather than pretending a clean factorization exists when it doesn't.
- How it works: the roots r₁ and r₂ are found using the quadratic formula, then plugged directly into the factored form a(x − r₁)(x − r₂), which always expands back to the original expression.
- Not always "nice" whole numbers: unlike textbook examples designed to factor neatly, real-world quadratics often have irrational roots — this calculator still factors them correctly using decimal values.
- Not factorable when the discriminant is negative: if the roots are complex (not real numbers), the expression can't be factored using only real numbers, and this calculator flags that case explicitly.
Can I verify the factored form is correct?
Yes — multiply the two factors back together (FOIL) and it should expand exactly to your original ax² + bx + c expression; this is a useful check whenever you factor by hand.
Why does the factored form sometimes have decimals instead of whole numbers?
Many quadratics simply don't have rational roots — the roots can be irrational (like involving √2) even though the original coefficients are whole numbers, so the factored form reflects that with decimal approximations.
Quadratic Factoring Calculator
Enter the coefficients a, b, and c from your quadratic expression. This finds the roots using the quadratic formula and writes the fully factored form a(x − r₁)(x − r₂) — or tells you clearly when the expression can't be factored over the real numbers.
The roots r₁ and r₂ are found using the quadratic formula and then plugged directly into the factored form, which always expands back to exactly the original expression — a useful way to check factoring work by hand. Unlike textbook examples designed to factor into neat whole numbers, real-world quadratics often have irrational or complex roots that don't factor "nicely" at all, and this calculator reports those cases plainly rather than pretending a clean factorization exists when it doesn't.
- How it works: the roots r₁ and r₂ are found using the quadratic formula, then plugged directly into the factored form a(x − r₁)(x − r₂), which always expands back to the original expression.
- Not always "nice" whole numbers: unlike textbook examples designed to factor neatly, real-world quadratics often have irrational roots — this calculator still factors them correctly using decimal values.
- Not factorable when the discriminant is negative: if the roots are complex (not real numbers), the expression can't be factored using only real numbers, and this calculator flags that case explicitly.
Can I verify the factored form is correct?
Yes — multiply the two factors back together (FOIL) and it should expand exactly to your original ax² + bx + c expression; this is a useful check whenever you factor by hand.
Why does the factored form sometimes have decimals instead of whole numbers?
Many quadratics simply don't have rational roots — the roots can be irrational (like involving √2) even though the original coefficients are whole numbers, so the factored form reflects that with decimal approximations.