Quadratic Equation Solver (Free Step-by-Step Tool)

Use this free quadratic equation solver to solve any equation in the form ax² + bx + c = 0. Enter the three coefficients and instantly get the roots (real or complex), the discriminant, the vertex and axis of symmetry, the factored form, a live parabola graph, and a clear step-by-step solution.

How to use the quadratic equation solver

1. Write your equation in standard form, ax² + bx + c = 0, so the right-hand side is zero.
2. Enter a, b, and c in the three boxes. Decimals and negatives are fine; a cannot be 0 (that would make the equation linear).
3. Press Solve. You will see the roots, discriminant, vertex, axis of symmetry, y-intercept, factored form, a graph, and the full working.
4. Share or save: the page link updates with your coefficients (for example ?a=2&b=5&c=-3), and the Copy results button copies everything to your clipboard.

Worked examples

Two real roots: For 2x² + 5x − 3 = 0, the discriminant is 5² − 4(2)(−3) = 49, so x = 1/2 or x = −3.

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One repeated (double) root: For x² − 4x + 4 = 0, the discriminant is 0, so x = 2 is a double root and the parabola just touches the x-axis.

Complex roots: For x² + x + 1 = 0, the discriminant is −3, so there are no real roots; instead x = −0.5 ± (√3/2)i.

What the solver shows you

• Roots: the solutions of the equation, shown as exact fractions when possible.
• Discriminant (b² − 4ac): tells you whether the roots are two real, one repeated, or two complex.
• Vertex and axis of symmetry: the turning point of the parabola and its line of symmetry.
• Y-intercept and factored form: where the curve meets the y-axis, and the equation rewritten as a product of factors.
• Graph: the parabola with roots and vertex marked.

Frequently asked questions

How do I use the quadratic equation solver?

Enter the coefficients a, b, and c from your equation in standard form ax² + bx + c = 0, then press Solve. The tool shows the roots, discriminant, vertex, axis of symmetry, y-intercept, factored form, a graph, and the full step-by-step solution.

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What does it mean when the discriminant is negative?

When b² − 4ac is less than zero, the equation has no real solutions. The solver returns two complex conjugate roots of the form p + qi and p − qi, and the parabola does not cross the x-axis.

Can a quadratic have exactly one solution?

Yes. When the discriminant equals zero, both roots are identical (a double root) and the parabola touches the x-axis at exactly one point, its vertex.

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Is this quadratic equation solver free?

Yes. The Effortless Math quadratic equation solver is completely free, works in your browser, and requires no sign-up.

Want the full lesson with methods, worked examples, and a free worksheet? Read How to Solve a Quadratic Equation.

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