Factor a quadratic into brackets where it factors rationally, and break any integer into its prime factors.
Understanding the factoring calculation
A quadratic factors neatly over the rationals only when its discriminant is a perfect square. The calculator checks that first, then produces the bracketed form; when it is not a perfect square it reports the exact roots instead of forcing an ugly factorisation.
The vertex and axis of symmetry come free from the same coefficients, and the integer factoriser handles the separate but related task of prime decomposition.
Worked example
On the values this calculator opens with, the factored form is (2x + 1)(x + 3). Underneath, Discriminant comes out at 25 and Roots at x = -0.5 , x = -3. Change any field and every figure updates as you type.
Getting your answer
- Enter the coefficients a, b and c.
- Read the factored form or the roots.
- Enter any integer to see its prime factorisation.
The formula
discriminant Δ = b² − 4ac
roots = (−b ± √Δ) ÷ 2a · vertex at x = −b ÷ 2a
Frequently asked questions
When does a quadratic factor nicely?
When the discriminant is a perfect square. Otherwise the roots are irrational or complex and brackets with whole numbers are impossible.
What does a negative discriminant mean?
No real roots — the parabola never crosses the x-axis. The roots are a complex conjugate pair.
How does prime factorisation work?
By dividing out the smallest prime repeatedly until nothing is left, which is trial division up to the square root.