How to use

The quadratic equation calculator solves ax²+bx+c=0 when a≠0. It first evaluates the discriminant to identify the root type, then reports real or complex roots together with the parabola vertex, axis of symmetry and y-intercept.

Rewrite the equation as ax²+bx+c=0.
Enter a, b and c; use zero for a missing term, but a itself cannot be zero.
Calculate and review the discriminant, roots, vertex and symmetry axis.

How it is calculated

Δ = b²−4ac; roots x = (−b ± √Δ) ÷ (2a); vertex h = −b/(2a), k = f(h)

Example

Example: x²−5x+6=0 has a=1, b=−5 and c=6. The discriminant is 1, so the roots are 2 and 3. The vertex is (2.5, −0.25) and the symmetry axis is x=2.5.

Important notes

If Δ>0 there are two distinct real roots; Δ=0 gives one repeated real root; Δ<0 gives a complex conjugate pair. For real roots the calculator uses a numerically more stable variant of the quadratic formula to reduce cancellation error in some cases, but browser floating-point precision still applies and this is not a symbolic algebra system.

Frequently asked questions

What if a equals zero?

Then the equation is not quadratic, so this calculator rejects it instead of applying the wrong formula.

What does a negative discriminant mean?

There are no real roots; the solutions form a complex conjugate pair.

Is the vertex the same as a root?

No. The vertex is the parabola’s turning point, while roots are x-values where y=0.

Is this calculator free?

Yes. It is free to use and requires no account.

Quadratic Equation Calculator