What is the quadratic formula?

The quadratic formula states that for ax² + bx + c = 0 (a ≠ 0), the solutions are x = (−b ± √(b² − 4ac)) / (2a). The expression under the square root, b² − 4ac, is called the discriminant.

What does the discriminant tell me?

If the discriminant is positive, the equation has two distinct real roots. If it is zero, there is exactly one repeated real root. If it is negative, the roots are complex conjugates of the form p ± qi.

No. If a = 0, the equation reduces to bx + c = 0, which is linear, not quadratic. The solver will display an error in that case.

What is the vertex of a parabola?

The vertex is the highest or lowest point of the parabola. Its x-coordinate is h = −b / (2a) and its y-coordinate is k = a·h² + b·h + c. The axis of symmetry is the vertical line x = h.

How are complex roots displayed?

Complex roots appear in conjugate pairs as p + qi and p − qi, where p = −b/(2a) and q = √|Δ|/(2a). The imaginary unit i satisfies i² = −1.

Quadratic Equation Solver

Name: Quadratic Equation Solver
Availability: InStock
Author: Nham Vu

Enter a, b, c and instantly solve ax² + bx + c = 0. Shows roots, discriminant, vertex, and steps.

Coefficients

ax² + bx + c = 0

equation preview

a (x²)

b (x)

c (const)

Enter coefficients a, b, and c on the left, then click Solve.

Coefficient a must not be zero.

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Summary

Enter a, b, c and instantly solve ax² + bx + c = 0. Shows roots, discriminant, vertex, and steps.

How it works

Enter the coefficient a (the x² term) — must not be zero.
Enter the coefficient b (the x term) — can be any real number, including zero.
Enter the coefficient c (the constant term) — can be any real number, including zero.
Click "Solve" or press Tab to move between fields; results update automatically.
Click any root value to copy it to your clipboard.

Use cases

Quickly check homework or exam answers for quadratic equations.
Find the roots of a parabola when graphing or analyzing functions.
Determine whether a quadratic has real solutions before solving by hand.
Calculate the vertex and axis of symmetry for optimization problems in physics or economics.

Frequently Asked Questions

Last updated: 2026-06-09 · Reviewed by Nham Vu