Quadratic Formula Calculator

Solve any quadratic equation ax² + bx + c = 0 for x — this quadratic equation solver finds real or complex roots and shows the discriminant and every formula step.

x2 + x + = 0
Roots (x) —

How to Solve a Quadratic Equation

Enter the coefficients from your equation in standard form to see the roots and the working.

1

Write your equation in standard form

Rearrange your equation to look like ax² + bx + c = 0, then read off the coefficients a, b, and c (a cannot be 0, or it isn't a quadratic).

2

The calculator applies the quadratic formula

It computes the discriminant b² - 4ac, then applies x = (-b ± √(b² - 4ac)) / (2a) to find the roots — real if the discriminant is zero or positive, complex if it's negative.

3

Read the roots and steps

Both solutions for x appear at the top, with the discriminant and every formula step listed below. Solving two equations together instead? The System of Equations Solver handles a linear pair of unknowns.

Frequently Asked Questions

What is the quadratic formula?
The quadratic formula solves any equation of the form ax² + bx + c = 0 (where a is not 0) for x: x = (-b ± √(b² - 4ac)) / (2a). Plug in the coefficients a, b, and c, compute the discriminant b² - 4ac under the square root, and the ± gives you the two possible roots. This calculator does that arithmetic for you and shows each step.
How do you use this quadratic equation solver?
Write your equation in standard form ax² + bx + c = 0, then enter the three coefficients a, b and c in the fields above. For x² + 5x + 6 = 0 that is a = 1, b = 5 and c = 6, and the solver returns x = −2 and x = −3. It computes the discriminant first and then lists every step of the quadratic formula, so you can check the working rather than just the answer. The coefficient a must not be zero, because the equation would then be linear rather than quadratic.
What does the discriminant tell you?
The discriminant is the b² - 4ac part under the square root in the quadratic formula, and its sign tells you what kind of roots to expect before you finish solving. A positive discriminant means two distinct real roots, exactly zero means one repeated real root where the parabola just touches the x-axis, and a negative value means no real roots but two complex-conjugate ones. For x² + 5x + 6 the discriminant is 25 - 24 = 1, so there are two real roots. A discriminant that is a perfect square is also a useful signal that the quadratic factors over the rationals.
What happens when the discriminant is negative?
A negative discriminant means the square root in the quadratic formula is a square root of a negative number, so the two roots come out as a complex-conjugate pair in the form p + qi and p - qi rather than real numbers. Graphically, this means the parabola never crosses the x-axis. This calculator still solves those cases and reports both complex roots.

Related algebra calculators

Factoring Calculator

Factor a polynomial over the rationals — GCF, difference of squares, and rational-root steps.

Algebra

→

Polynomial Calculator

Add, subtract, multiply, or divide two polynomials, with long division and a remainder.

Algebra

→

Slope Calculator

Slope from two points, plus the angle, y = mx + b, distance, and midpoint.

Algebra

→

System of Equations Solver

Solve two linear equations in two unknowns with Cramer’s rule, step by step.

Algebra

→

All Math Calculators

Algebra, matrices, statistics and geometry with step-by-step solutions.

Math

→