Quadratic Equation Solver
a x² + b x + c = 0
Roots
x = 3 or x = 2
Discriminant
1
Vertex
(2.5, -0.25)
Axis of symmetry
x = 2.5
About Quadratic Equation Solver
Enter a, b, and c and solve ax² + bx + c = 0 instantly — real or complex roots, the discriminant, the vertex, and the axis of symmetry, all at once.
The quadratic formula gives the roots of any equation in the form ax² + bx + c = 0, but working it through by hand — especially when the discriminant comes out negative, meaning the roots are complex numbers — is a common place to make an arithmetic slip. This computes it directly and shows every intermediate value, not just the final answer.
The discriminant (b² − 4ac) determines the shape of the answer before you even compute the roots: positive means two distinct real roots, zero means one repeated real root, negative means a pair of complex conjugate roots. All three cases are handled and labeled clearly rather than one being treated as an error.
The vertex and axis of symmetry describe the parabola itself, not just where it crosses zero — useful for graphing the equation or finding its minimum or maximum point, which the roots alone don't tell you.
What you can use it for
- Solve a quadratic equation for a homework or test problem.
- Find the vertex of a parabola for graphing.
- Check whether an equation has real or complex roots before solving it by hand.
- Verify a factored form against the quadratic formula's result.
- Find where a parabola crosses the x-axis.
How it works, step by step
- 1
Enter a, b, and c
The coefficients of ax² + bx + c = 0.
- 2
Read the roots
Real, repeated, or complex, depending on the discriminant.
- 3
Check the vertex and axis of symmetry
Both are computed from the same coefficients.
Frequently asked questions
What does it mean if the roots are complex?
The parabola never crosses the x-axis — it stays entirely above or entirely below it. The roots still exist mathematically, expressed as a real part plus or minus an imaginary part.
Why can't "a" be zero?
With a = 0 the x² term disappears and the equation becomes linear (bx + c = 0), which has a different, simpler solution method — this tool is specifically for the quadratic case.
What's the discriminant used for?
It's the part of the quadratic formula under the square root (b² − 4ac). Its sign alone tells you whether the roots are two real numbers, one repeated real number, or a complex pair, before you finish the calculation.
Is my equation sent anywhere?
No. Everything is solved entirely in your browser.