Quadratic Equation Solver
Solve ax² + bx + c = 0 instantly: real or complex roots, discriminant, vertex, factored form and full step-by-step working with a parabola graph.
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- Root x₁
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- Root x₂
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- Discriminant (b² − 4ac)
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- Quadratic formula
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- Vertex
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- Axis of symmetry
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- Sum of roots
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- Product of roots
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- Factored form
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Step by step
Graph of the parabola
How it works
Type the three coefficients of ax² + bx + c = 0 and the solver applies the quadratic formula x = (−b ± √(b² − 4ac)) / 2a as you type. It first computes the discriminant b² − 4ac, because its sign decides everything: positive gives two distinct real roots, zero gives one repeated root where the parabola just touches the x-axis, and negative gives a pair of complex conjugate roots of the form p ± qi.
Alongside the roots you get the vertex, the axis of symmetry x = −b/2a, the sum (−b/a) and product (c/a) of the roots, the factored form and a sketch of the parabola so you can see where it crosses the axis. Every step of the working is listed so you can check your own homework rather than just copy an answer. All the arithmetic runs in your browser in plain JavaScript — nothing is uploaded, there is no signup and no equation ever leaves your device.
Frequently asked questions
How do you solve a quadratic equation step by step?
Write the equation in the form ax² + bx + c = 0, identify the three coefficients, then compute the discriminant b² − 4ac. Take its square root and substitute everything into the quadratic formula x = (−b ± √(b² − 4ac)) / 2a, which gives the two roots. This solver prints each of those steps with your own numbers filled in — the coefficients, the discriminant arithmetic, the substituted formula and the final roots — so you can check your working rather than just copy an answer.
What does a negative discriminant mean?
The discriminant b² − 4ac decides the nature of the roots. If it is positive the parabola crosses the x-axis twice and there are two distinct real roots; if it is exactly zero the parabola just touches the axis and there is one repeated root; if it is negative the parabola never reaches the x-axis and the two roots are complex conjugates of the form p ± qi. The solver labels which case you are in and still shows the complex roots when the discriminant is negative.
Why can't coefficient a be zero?
If a is 0 the x² term disappears and bx + c = 0 is a linear equation, not a quadratic — the quadratic formula would divide by 2a = 0. The tool shows a friendly message instead of a wrong answer, so change a to any non-zero value (it can be negative or a decimal) to get the roots. Everything is calculated locally in your browser, so no equation is ever uploaded.