Quadratic Equation Solver

Enter the coefficients a, b and c to solve ax² + bx + c = 0. The solver uses the quadratic formula to find both roots — real or complex — and also reports the discriminant and the parabola's vertex.

Enter a, b and c, then press Solve.

The quadratic formula

A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0. Its solutions (roots) are given by the quadratic formula, which works for every quadratic — even ones that don't factor neatly.

x = ( −b ± √(b² − 4ac) ) ÷ 2a

The expression under the square root, b² − 4ac, is the discriminant. Its sign tells you the nature of the roots before you finish solving: positive means two distinct real roots, zero means one repeated real root, and negative means two complex-conjugate roots.

Worked example

Solve x² − 3x + 2 = 0, so a = 1, b = −3, c = 2:

Discriminant: (−3)² − 4·1·2 = 9 − 8 = 1.
Roots: x = (3 ± √1) ÷ 2 = (3 ± 1) ÷ 2.
Result: x = 2 and x = 1 — two real roots.

Discriminant and vertex

Beyond the roots, the solver shows the discriminant and the parabola's vertex. The vertex sits on the axis of symmetry at x = −b ÷ 2a, and its y-value is the equation evaluated there — the minimum of the parabola when a > 0, or the maximum when a < 0. When the discriminant is negative the roots are complex and written as p ± qi.

Tip: need the slope or intercept of a line instead? Use the slope calculator, or reduce fractions with the fraction calculator.

Frequently asked questions

What if a is 0?

Then the equation isn't quadratic — it's linear (bx + c = 0). The solver flags this and solves the linear case x = −c ÷ b instead, since the quadratic formula divides by a.

What does the discriminant tell me?

The discriminant b² − 4ac decides the roots: positive gives two distinct real roots, zero gives one repeated real root, and negative gives two complex-conjugate roots of the form p ± qi.

Can it give complex roots?

Yes. When b² − 4ac is negative the real square root doesn't exist, so the solver returns complex roots written as p ± qi, where q = √(4ac − b²) ÷ 2a.

What is the vertex used for?

The vertex is the turning point of the parabola y = ax² + bx + c, at x = −b ÷ 2a. It's the minimum value when a is positive and the maximum when a is negative — useful in optimization problems.

MB
Mustafa Bilgic · Editor, Calcool
Uses the standard quadratic formula with discriminant analysis for real and complex roots. Everything runs in your browser — nothing you enter is uploaded, logged or stored.

Related calculators

📐 Slope Calculator🔬 Scientific Calculator½ Fraction Calculator％ Percentage Calculator