Quadratic Equation Solver

Enter a, b and c to solve ax² + bx + c = 0 — real or complex roots, discriminant, vertex, and axis of symmetry.

ax² + bx + c = 0

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The Quadratic Formula

Any equation in the form ax² + bx + c = 0 (with a ≠ 0) can be solved using the quadratic formula: x = (−b ± √(b² − 4ac)) / (2a). The expression under the square root, b² − 4ac, is called the discriminant — its sign tells you what kind of roots to expect before you even finish solving.

  • Discriminant > 0 — two distinct real roots (the parabola crosses the x-axis twice)
  • Discriminant = 0 — one repeated real root (the parabola touches the x-axis once, at its vertex)
  • Discriminant < 0 — two complex roots (the parabola never touches the x-axis)

This is a general-purpose math tool for algebra and education — the math is universal and works the same everywhere.

Frequently Asked Questions

What is the quadratic formula?+

For ax² + bx + c = 0 (with a not equal to 0), the quadratic formula gives x = (−b ± √(b² − 4ac)) / (2a). This works for any quadratic equation, producing real or complex roots.

What does the discriminant tell you?+

The discriminant is b² − 4ac. If it's positive, the equation has two distinct real roots. If it's zero, there's exactly one repeated real root. If it's negative, the roots are complex (not real) numbers.

What if there are no real solutions?+

When the discriminant is negative, the parabola never crosses the x-axis, so there are no real x-intercepts. The two solutions still exist as complex numbers, written in the form p ± qi, where i is the imaginary unit.

How do you find the vertex of a parabola?+

For y = ax² + bx + c, the vertex's x-coordinate is h = −b / (2a). Substitute h back into the equation to get the y-coordinate, k. The vertex is the parabola's minimum point if a is positive, or its maximum point if a is negative.

What real-world problems use quadratic equations?+

Quadratic equations show up anywhere something follows a curved, accelerating pattern — projectile motion (the height of a thrown ball over time), maximizing area or profit, and satellite dish or bridge-arch shapes, among many others.