Quadratic Equation Solver
Solve ax² + bx + c = 0 with step-by-step working and parabola sketch.
(0, 0) = (−b/2a, f(−b/2a))(x − 0)² = 0x = (−b ± √(b² − 4ac)) / 2ax = (−(0) ± √((0)² − 4×1×0)) / (2×1)Δ = 0² − 4×1×0 = 0 − 0 = 0x₁ = (0 + √0) / 2 = 0 x₂ = (0 − √0) / 2 = 0Frequently Asked Questions
What is the quadratic formula?
The quadratic formula solves ax² + bx + c = 0: x = (−b ± √(b²−4ac)) ÷ 2a. The ± gives two solutions (roots). It's derived by completing the square and works for all quadratics (unlike factoring, which only works for factorable equations). Enter your a, b, c coefficients and the solver shows the full working.
What is the discriminant and what does it tell you?
The discriminant Δ = b² − 4ac determines the nature of the roots without solving. If Δ > 0: two distinct real roots (parabola crosses x-axis twice). If Δ = 0: one repeated real root (parabola just touches x-axis). If Δ < 0: two complex (imaginary) roots (parabola doesn't touch x-axis).
What is the vertex of the parabola?
The vertex is the turning point: maximum if a < 0 (opens down), minimum if a > 0 (opens up). Vertex coordinates: x = −b/(2a), y = f(−b/2a). The axis of symmetry is the vertical line x = −b/2a. The vertex also equals the point where the derivative of the quadratic is zero.
How do I factor a quadratic equation?
For ax² + bx + c, find two numbers that multiply to ac and add to b. Example: x² + 5x + 6 = (x+2)(x+3) because 2×3=6 and 2+3=5. Not all quadratics factor over integers — that's why the quadratic formula is the universal method. The factored form is shown automatically if it exists.