Quadratic Equation Solver

Roots of ax² + bx + c = 0, real or complex.

Overview

Solves quadratic equations of the form ax² + bx + c = 0, calculating real or complex roots, the discriminant, vertex coordinates, and axis of symmetry.

Formula & Calculation Method

Roots x = [-b ± sqrt(b² - 4ac)] / (2a) | Discriminant Delta = b² - 4ac

If Δ > 0: two distinct real roots. If Δ = 0: one repeated real root. If Δ < 0: two complex conjugate roots (u ± vi).

Worked Example: Solving 2x² - 4x - 6 = 0

Inputs: a = 2 | b = -4 | c = -6

  • Discriminant Δ = (-4)² - 4(2)(-6) = 16 - (-48) = 64 (Positive, 2 real roots)
  • Square root of discriminant = sqrt(64) = 8
  • Root 1: [-(-4) + 8] / (2 × 2) = (4 + 8) / 4 = 12 / 4 = 3.0
  • Root 2: [-(-4) - 8] / (2 × 2) = (4 - 8) / 4 = -4 / 4 = -1.0

Summary: Roots = x = 3 and x = -1 | Discriminant = 64 | Parabola Vertex = (1, -8)

Common Use Cases

  • Projectile Trajectory Modeling: Calculate flight duration, maximum height, and landing point for launched objects.
  • Profit Maximization: Find optimal product pricing coordinates where quadratic revenue curves peak.
  • Civil Engineering: Design parabolic suspension bridge curves and architectural archways.

Frequently Asked Questions

What does the discriminant tell you?

The discriminant (b² - 4ac) indicates the nature of roots: > 0 means 2 real roots, = 0 means 1 real root, < 0 means 2 complex roots.

What if the discriminant is negative?

The parabola does not cross the x-axis, and roots are complex numbers involving the imaginary unit i = sqrt(-1).

How do I find the vertex of a quadratic function?

The x-coordinate is -b / (2a); plug this x value back into the quadratic equation to find the corresponding y-coordinate.

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