Quadratic Equation Solver
Solve ax² + bx + c = 0 for real or complex roots, with the discriminant.
A quadratic equation has the form ax² + bx + c = 0, and this calculator solves it using the quadratic formula, x = (−b ± √(b² − 4ac)) / 2a. Enter the coefficients a, b and c and it returns the roots, the discriminant, and the coordinates of the parabola's vertex.
The discriminant, b² − 4ac, tells you what kind of roots to expect. A positive value gives two distinct real roots, zero gives a single repeated root, and a negative value gives a pair of complex conjugate roots written as real ± imaginary i. The coefficient a must not be zero, otherwise the equation is linear rather than quadratic.
How to use it
- 1Enter the coefficient a (it cannot be 0).
- 2Enter the coefficients b and c.
- 3Read the roots, the discriminant, and the vertex.
- 4Change any coefficient to see how the roots and shape of the parabola respond.
Frequently asked questions
What is the quadratic formula?+
x = (−b ± √(b² − 4ac)) / 2a. It gives the roots of any equation in the form ax² + bx + c = 0, as long as a is not zero.
What does the discriminant tell me?+
The discriminant is b² − 4ac. If it is positive there are two real roots, if it is zero there is one repeated root, and if it is negative the two roots are complex conjugates.
Why must a not be zero?+
If a is 0 there is no x² term, so the equation is linear, not quadratic, and the quadratic formula divides by 2a — which would be division by zero. The calculator shows a note instead.
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