Quadratic Equation Solver - ax² + bx + c = 0 (Real & Complex Roots)

Solve any quadratic ax² + bx + c = 0 - real or complex roots, discriminant, and vertex

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Solve ax² + bx + c = 0
x₁
2
x₂
1
Discriminant (b²−4ac)
1
Vertex x
1.5
Vertex y
-0.25
Quadratic formula: x = (−b ± √(b²−4ac)) ÷ 2a. Discriminant sign tells you: >0 two real roots, =0 one repeated, <0 two complex conjugate roots.
Estimates only. Results are generated by generalised formulas from the inputs you provide and are for informational purposes only. They are not financial, tax, legal, medical, or professional advice, and do not account for your individual circumstances, jurisdiction, taxes, fees, or health conditions. Verify important figures with a qualified professional. By using this tool you agree to our Terms & Disclaimer.

How it works

  1. 1
    Enter a, b, c

    The three coefficients of ax² + bx + c = 0. Any real numbers, positive or negative.

  2. 2
    Read the roots

    Two real, one repeated, or two complex conjugate roots depending on the discriminant.

  3. 3
    See vertex and discriminant

    Vertex tells you the parabola&#39;s minimum or maximum point. Discriminant (b²−4ac) sign classifies the roots.

About Quadratic Equation Solver - ax² + bx + c = 0 (Real & Complex Roots)

Free quadratic equation solver. Enter the coefficients a, b, and c of ax² + bx + c = 0 to see the roots (real or complex), the discriminant, and the vertex coordinates of the parabola. Handles all three discriminant cases: two real roots, one repeated root, or two complex conjugate roots. Runs entirely in your browser. Quadratic Equation Solver - ax² + bx + c = 0 (Real & Complex Roots) on 712 Tools runs entirely inside your browser using modern JavaScript APIs - no server ever sees your data. That means instant results, complete privacy, and no upload limits.

Whether you need to solve any quadratic ax² + bx + c = 0 - real or complex roots, discriminant, and vertex for a debugging session, a quick sanity check, or a production incident, this tool is free to use as many times as you need. There are no watermarks, no sign-up, and no ads inside the tool itself.

Frequently asked questions

What is the quadratic formula?

x = (−b ± √(b²−4ac)) ÷ 2a. The ± gives two solutions when the discriminant is positive.

What happens if a = 0?

The equation is linear, not quadratic. The tool solves x = −c ÷ b when b is non-zero.

What are complex roots?

When b² − 4ac < 0 the square root is imaginary, giving two complex conjugate solutions of the form p ± qi.