Quadratic Equation Solver
Solve ax² + bx + c = 0. Returns both real roots using the quadratic formula. Enter 0 for missing terms.
Results
What is it?
A quadratic equation is any polynomial of the form ax² + bx + c = 0. The quadratic formula x = (−b ± √(b²−4ac)) / 2a always yields the solutions. The discriminant Δ = b²−4ac determines the nature of the roots: if Δ > 0 there are two distinct real roots, Δ = 0 gives one repeated root, and Δ < 0 means the roots are complex (imaginary).
How to use
Enter coefficients a, b, and c from your equation. If a term is missing (e.g., no x term), enter 0 for b. The calculator returns the discriminant, both roots, and the vertex x-coordinate. When Δ < 0, Root x₁ shows the real part and Root x₂ shows the magnitude of the imaginary part (±imaginary).
Example scenario
For x² − 5x + 6 = 0 (a=1, b=−5, c=6): Δ = 25 − 24 = 1 > 0. Root x₁ = (5 + 1)/2 = 3, Root x₂ = (5 − 1)/2 = 2. The parabola crosses the x-axis at x=2 and x=3.
Pro tip
The vertex form of a quadratic is a(x − h)² + k, where h = −b/2a (the vertex x shown here) and k = c − b²/4a. This form makes it easy to see the minimum/maximum of the parabola. Quadratics appear in projectile motion, economics (profit maximisation), and optics.