Quadratic Equation Solver
Solve any quadratic equation ax² + bx + c = 0. Enter the three coefficients to get the step-by-step solution, discriminant analysis, exact or approximate roots, vertex, axis of symmetry, and a downloadable parabola graph.
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About the quadratic formula
The discriminant Δ = b² − 4ac
The discriminant determines the nature of the roots without solving the equation. If Δ > 0 there are two distinct real roots; if Δ = 0 the equation has one repeated root; if Δ < 0 the roots are complex conjugates.
The quadratic formula
For ax² + bx + c = 0 with a ≠ 0, the roots are x = (−b ± √Δ) / (2a). This formula always works, even when factoring by inspection is difficult.
The parabola and its vertex
The graph of f(x) = ax² + bx + c is a parabola with vertex at (−b/2a, f(−b/2a)) and axis of symmetry at x = −b/2a.
Frequently asked questions
For ax² + bx + c = 0, the roots are x = (−b ± √(b²−4ac)) / (2a). The expression Δ = b²−4ac is called the discriminant.
Δ > 0: two distinct real roots. Δ = 0: one repeated real root. Δ < 0: two complex conjugate roots (no real solutions).
The vertex of y = ax² + bx + c is at x = −b/(2a) and y = c − b²/(4a). It is the minimum when a > 0 and the maximum when a < 0.
The vertical line x = −b/(2a) divides the parabola into two mirror-image halves. It always passes through the vertex.