Quadratic Equation Solver
This solver finds all roots of any quadratic equation of the form ax² + bx + c = 0. Using the quadratic formula, it calculates the discriminant (b² - 4ac) and determines whether the equation has real roots, a single repeated root, or complex roots. For real roots, it displays both the exact values and their decimal approximations. For complex roots, it shows them in a + bi format. The solver also displays the step-by-step working, making it useful for learning and verification. Quadratic equations are fundamental in algebra and have applications across mathematics, physics, engineering, and economics.
Quadratic formula and discriminant
Standard form: ax² + bx + c = 0
Discriminant: D = b² - 4ac
Quadratic formula: x = (-b ± sqrt(D)) / (2a)
If D > 0: two distinct real roots
If D = 0: one repeated real root
If D < 0: two complex conjugate roots
Quadratic equations: frequently asked questions
What is a quadratic equation?
A quadratic equation is a polynomial equation of degree 2, written in the standard form ax² + bx + c = 0, where a, b, and c are constants and a is not zero. Quadratic equations have at most two solutions, which can be real or complex.
What is the quadratic formula?
The quadratic formula solves any quadratic equation: x = (-b ± sqrt(b² - 4ac)) / (2a). The expression under the square root, b² - 4ac, is called the discriminant. It determines the nature and number of the solutions.
What is the discriminant?
The discriminant (D = b² - 4ac) determines the nature of the roots. If D > 0, there are two distinct real roots. If D = 0, there is one repeated real root. If D < 0, there are two complex conjugate roots.
What are complex roots?
Complex roots occur when the discriminant is negative. They are expressed in the form a + bi, where i is the imaginary unit (sqrt(-1)). For example, 2 + 3i and 2 - 3i are complex conjugate roots.
How do I use this solver?
Enter the coefficients a, b, and c from your quadratic equation ax² + bx + c = 0. The solver calculates the discriminant and displays the roots. If the roots are real, it shows their exact values. If complex, it shows them in a ± bi format.
Methodology
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.