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Quadratic Equation Solver

Solve ax² + bx + c = 0 with the quadratic formula, reporting the discriminant and both roots, including complex roots.

Discriminant (b² − 4ac)
1
Nature
Two distinct real roots
Root 1
3
Root 2
2

y = ax² + bx + c

The curve crosses the x-axis at each real root.

How it works

The quadratic formula solves ax² + bx + c = 0 as x = (−b ± √(b² − 4ac)) ÷ 2a. The discriminant, b² − 4ac, determines what kind of roots you get: positive means two distinct real roots, zero means one repeated real root, and negative means a pair of complex conjugate roots (since you'd be taking the square root of a negative number).

Coefficient a can't be zero — if it were, the x² term would vanish and the equation would just be linear, not quadratic.

The plot below shows the actual curve y = ax² + bx + c, centered on its vertex, with a dot marking each real root where the curve crosses the x-axis — for a complex pair, the curve simply won't touch the axis at all, which is a more direct way to see what a negative discriminant means than the number alone.

FAQ

What does it mean if the roots are complex?

A negative discriminant means the parabola y = ax² + bx + c never crosses the x-axis — there's no real number x that makes the equation zero. The two complex roots still exist mathematically as a conjugate pair (same real part, opposite-sign imaginary part), which is what's shown.

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