Quadratic Equation Calculator

A quadratic equation has the standard form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. Its graph is a parabola, and the solutions (or roots) are the x-values where the parabola crosses the x-axis.

This calculator solves any quadratic using the quadratic formula and also returns the discriminant, the vertex, the axis of symmetry, and whether the parabola opens upward or downward. Together those values give you a complete picture of the curve, not just where it equals zero.

Solve 2x² − 4x − 6 = 0. Here a = 2, b = −4, c = −6.

• Discriminant: Δ = (−4)² − 4(2)(−6) = 16 + 48 = 64 (positive → two real roots)
• √64 = 8
• x = (4 + 8) / 4 = 3
• x = (4 − 8) / 4 = −1
• Axis of symmetry: x = 4 / 4 = 1
• Vertex: y = 2(1)² − 4(1) − 6 = −8, so vertex is (1, −8)

Quick sanity check by factoring: 2x² − 4x − 6 = 2(x − 3)(x + 1), confirming roots x = 3 and x = −1.

• Solving polynomial roots in algebra and pre-calculus homework.
• Projectile motion — find when a thrown object hits the ground (h(t) = −½gt² + v₀t + h₀ = 0).
• Optimizing area or revenue — the vertex of a downward parabola gives the maximum.
• Engineering and physics — analyzing oscillations, parabolic trajectories, and lensing.
• Curve fitting — once you have a quadratic regression, the formula gives you intercepts and the turning point.

• Forgetting the ± sign. The formula yields two roots; missing one is a common test mistake.
• Sign errors on b. The formula uses −b, so b = −5 contributes +5, not −5.
• Treating a negative discriminant as "no solution." There are still solutions — they're complex numbers.
• Dividing only the radical by 2a. Both −b and ±√Δ must be divided by 2a; parentheses matter.
• Confusing the vertex x-coordinate with a root. The vertex is the turning point; roots are where y = 0.