Quadratic Equation Solver

What this tool does

The Quadratic Equation Solver finds both roots of any quadratic, including complex roots when the discriminant is negative, and writes irrational roots exactly in simplified surd form. It shows each step of the quadratic formula, the vertex and factored forms, and a graph of the parabola.

A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0. Its solutions come from the quadratic formula x = (−b ± √(b² − 4ac)) ÷ 2a, and the expression under the root, the discriminant D = b² − 4ac, decides what kind of roots you get: two real roots when D > 0, one repeated root when D = 0, and a pair of complex conjugates when D < 0.

This solver walks through those steps with your coefficients, so you can compare them with your own working. Roots are computed with a numerically stable form of the formula that avoids losing precision when b² is much larger than 4ac. With whole-number coefficients, irrational roots are also written exactly, for example x = 1 ± √3. The vertex form, axis of symmetry, Vieta’s sum and product of roots and a plotted parabola complete the picture. Everything is calculated in your browser.

How to use Quadratic Equation Solver

1. Rearrange your equation so one side is zero, for example 2x² = 4x + 6 becomes 2x² − 4x − 6 = 0.
2. Enter the coefficients a, b and c. Use negative numbers for minus signs and 0 for missing terms.
3. Read the roots x₁ and x₂ and the nature of the roots.
4. Follow the numbered steps below the results to see the discriminant and the formula applied.
5. Use the vertex, vertex form and graph to understand the shape of the parabola; a positive a opens upwards.

Tips for better results

• If a = 0 the equation is linear, and the solver returns the single solution x = −c ÷ b.
• Check roots with Vieta’s formulas: they must add up to −b/a and multiply to c/a.
• A rational discriminant square with integer coefficients means the quadratic factorises neatly over the integers or rationals.
• The vertex gives the maximum or minimum value, which is what optimisation word problems usually ask for.

Common use cases

• Checking algebra homework and exam practice.
• Projectile motion problems where height is a quadratic function of time.
• Finding break-even points when revenue and cost curves are quadratic.
• Locating the maximum area or profit in optimisation questions.

Common mistakes to avoid

• Forgetting to move all terms to one side before reading off a, b and c.
• Dropping the sign of b: for x² − 5x + 6 = 0, b is −5, not 5.
• Dividing only part of the numerator by 2a; the whole of −b ± √D is divided.