Quadratic Equation Solver

About the Quadratic Equation Solver

This solves any quadratic equation of the form ax² + bx + c = 0, giving the roots (the values of x that make it zero), the discriminant, the nature of the roots, and the vertex of the parabola. It's a genuine help for algebra homework, physics problems, and anyone needing to solve or check a quadratic quickly — including the cases where the answers are complex numbers.

How it works — the quadratic formula

It uses the quadratic formula:

x = ( −b ± √(b² − 4ac) ) ⁄ 2a

The part under the root, b² − 4ac, is the discriminant, and it decides the nature of the roots:

• > 0 → two distinct real roots.
• = 0 → one repeated real root.
• < 0 → two complex roots (a real part ± an imaginary part), which the tool shows in p ± qi form.

It also gives the vertex at x = −b⁄2a, the turning point of the parabola.

Assumptions and behaviour

• The equation must be a quadratic, so a can't be 0 (that would make it linear) — the tool flags this.
• Complex roots are fully supported and shown when the discriminant is negative.
• The vertex (x, y) is the minimum or maximum point of the parabola.
• Coefficients can be decimals or negatives; results are rounded for display.

Limitations

• It solves quadratics only — second-degree equations. For linear (a = 0) or higher-degree equations (cubics, quartics), it isn't the right tool.
• Results are numeric, not symbolic — it gives decimal roots, not exact surds like (3 + √5)/2.
• Floating-point arithmetic can introduce tiny rounding errors, so a discriminant that should be exactly 0 may display as a minuscule value, occasionally affecting which case is reported at the boundary.
• The equation must already be in standard form ax² + bx + c = 0; rearrange first if needed.

Privacy

The calculation runs entirely in your browser. Nothing is uploaded or stored.