Quadratic Equation Solver
Solve ax² + bx + c = 0 with the quadratic formula — real or complex roots, plus the discriminant.
How it works
x = (−b ± √(b² − 4ac)) ÷ 2a
The quadratic formula solves any equation of the form ax² + bx + c = 0. The discriminant b² − 4ac decides everything before you finish: positive means two real roots, zero means one repeated root, negative means a pair of complex conjugate roots of the form p ± qi.
Worked example
x² − 3x − 10 = 0 has a = 1, b = −3, c = −10. Discriminant = 9 − 4(1)(−10) = 49, and √49 = 7. So x = (3 ± 7) ÷ 2, giving x₁ = 5 and x₂ = −2. Check: (x − 5)(x + 2) = x² − 3x − 10.
Frequently asked questions
What does a complex root mean?
It means the parabola never crosses the x-axis. Complex roots always come in mirror pairs like 2 + 3i and 2 − 3i, and they are meaningful answers in physics and engineering, not errors.
Can I solve a quadratic without the formula?
Often. Factoring works when the roots are nice whole numbers or fractions, and completing the square is how the formula itself is derived. The formula is just the method that never fails.
What do the roots look like on a graph?
They are the x-intercepts of the parabola y = ax² + bx + c. The vertex sits exactly halfway between them at x = −b ÷ 2a.
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This calculator is for educational purposes. Double-check important results before acting on them.