Quadratic Equation Solver
Solve ax² + bx + c = 0 with the quadratic formula: discriminant, real or complex roots, vertex and factored form. Free and private.
🔒 Runs entirely in your browser — nothing is uploadedThe quadratic formula
A quadratic equation has the form ax² + bx + c = 0 where a is not zero. Its solutions, called roots or zeros, are given by the quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a. The expression under the square root, b² − 4ac, is the discriminant. It decides what kind of answer you get before you even take the root, which makes it a quick check when you solve homework or analyze a model.
Reading the discriminant
When the discriminant is positive, the parabola crosses the x-axis twice and you get two different real roots. When it is zero, the parabola touches the axis at its vertex, giving one repeated root. When it is negative, the parabola never meets the axis, and the roots are a pair of complex conjugates written as p ± qi. For x² − 3x + 2 = 0, D = 9 − 8 = 1, so the roots are (3 ± 1) ÷ 2, which are 2 and 1. The factored form is then (x − 1)(x − 2).
Vertex and graph shape
The solver also reports the vertex, the highest or lowest point of the parabola, at x = −b ÷ 2a. If a is positive the parabola opens upward and the vertex is a minimum, and if a is negative it opens downward and the vertex is a maximum. This is useful for optimization, such as finding the maximum height of a thrown object or the price that maximizes revenue. All math runs in your browser, so your numbers stay private.
Tip: you can check any real root by substituting it back into the equation, which should give zero. The sum of the roots equals −b ÷ a and their product equals c ÷ a, two relationships (Vieta's formulas) that make a quick sanity check on your answer.
How to use
- Enter the coefficientsType a, b and c from your equation ax² + bx + c = 0.
- Check the discriminantIts sign tells you whether there are two real roots, one repeated root or two complex roots.
- Read the roots and vertexGet the roots, the factored form and the vertex of the parabola.