Quadratic Equation Solver
Solve ax² + bx + c = 0 and see the discriminant, the roots and the vertex.
About this tool
Enter the coefficients of a quadratic equation in the form ax² + bx + c = 0. The solver uses the quadratic formula, reports whether the roots are real, repeated or complex, and gives the turning point of the parabola. If a is 0 it solves the straight-line equation instead.
A quadratic equation has an x² term, like x² − 3x − 10 = 0, and its graph is a U-shaped curve called a parabola. Enter the three coefficients a, b and c to get both roots, the discriminant that tells you what kind of roots to expect, and the vertex, the curve’s lowest or highest point. It handles negative and decimal coefficients and gives complex roots when there are no real ones.
How to use the Quadratic Equation Solver
- Rearrange your equation so one side is 0, in the form ax² + bx + c = 0.
- Enter a, b and c, including any minus signs.
- Read the roots and the type of solution.
- Use the discriminant and vertex to check your working or sketch the graph.
Solving x² − 3x − 10 = 0
Here a = 1, b = −3 and c = −10. The discriminant is (−3)² − 4 × 1 × (−10) = 9 + 40 = 49, and √49 = 7. The quadratic formula gives x = (3 ± 7) ÷ 2, so x = 5 or x = −2. You can check by factorising: (x − 5)(x + 2) = 0. The vertex is at x = 3 ÷ 2 = 1.5, where y = 1.5² − 3 × 1.5 − 10 = −12.25.
The quadratic formula and the discriminant
Every quadratic can be solved with x = (−b ± √(b² − 4ac)) ÷ 2a. The part under the square root, b² − 4ac, is called the discriminant. If it is positive there are two different real roots and the parabola crosses the x-axis twice. If it is zero there is one repeated root and the curve just touches the axis at its vertex. If it is negative there are no real roots, because you cannot take the square root of a negative number in ordinary arithmetic; instead there are two complex roots of the form p ± qi. The vertex sits halfway between the roots at x = −b ÷ 2a. When a is positive the parabola opens upwards and the vertex is a minimum; when a is negative it opens downwards and the vertex is a maximum.
Frequently asked questions
What if my equation is not equal to zero?
Move every term to one side first. For example, x² = 4x + 5 becomes x² − 4x − 5 = 0, so a = 1, b = −4 and c = −5.
What does a negative discriminant mean?
There are no real solutions: the parabola never crosses the x-axis. The solver shows the two complex roots instead, written with i, the square root of −1.
Can a be 0?
Then the equation is not quadratic but linear, bx + c = 0. The solver detects this and gives the single answer x = −c ÷ b.
Why are some answers long decimals?
When the discriminant is not a perfect square the roots are irrational, such as (1 ± √5) ÷ 2. The solver shows them rounded to six decimal places.