Quadratic Equation Solver

How to use Quadratic Equation Solver

1. Write your equation as ax² + bx + c = 0.
2. Enter the coefficients a, b and c (decimals and negatives are fine).
3. Read the roots, the discriminant and the type of solution.
4. Use the steps and the vertex to sketch or check the parabola.

How the solver works

A quadratic equation has the form ax² + bx + c = 0 with a not equal to zero. The solver first computes the discriminant D = b² − 4ac. Its sign decides the kind of answer.

Worked example: x² − 3x + 2 = 0 has a = 1, b = −3, c = 2. D = 9 − 8 = 1, a positive number, so there are two real roots: (3 ± 1) / 2, giving x = 1 and x = 2. The vertex is at x = 3 / 2 = 1.5 and y = 2 − 9 / 4 = −0.25, so the parabola’s lowest point is (1.5, −0.25).

For x² + 2x + 5 = 0, D = 4 − 20 = −16, which is negative, so the roots are complex: −1 ± 2i.

Where quadratics appear

• Projectile motion, where height depends on the square of time.
• Area problems, such as finding the dimensions of a rectangle with a given area and perimeter.
• Break-even and optimisation in business, where profit follows a parabola.
• Physics and engineering formulas that contain a squared variable.

Notes and limits

Results are shown to about eight decimal places and may carry tiny floating-point differences. Very large or tiny coefficients (beyond about 10^150) are rejected. The tool finds roots only; it does not factor the equation symbolically. If a root looks odd, check by substituting it back into the equation. Everything runs locally in your browser.

Why use this quadratic equation solver tool

• Real and complex roots. Negative discriminants give complex conjugate roots written as p ± qi.
• Parabola details. Shows the vertex, axis of symmetry, y-intercept and whether the parabola opens upwards or downwards.
• Stable numerics. Uses a numerically stable form of the quadratic formula to avoid precision loss when b is large relative to a and c.
• Handles a = 0. If a is zero the equation is linear and the tool solves it as such instead of failing.

Frequently asked questions

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