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Quadratic Formula Calculator

Solve quadratic equations using the quadratic formula. Find real and complex roots, discriminant, vertex, and step-by-step solution.

ax² + bx + c = 0

The Quadratic Formula

For any equation ax² + bx + c = 0 where a ≠ 0, the solutions are given by:

x = (-b ± √(b² - 4ac)) / 2a

Discriminant (b² - 4ac)

The discriminant determines the nature of the roots:

  • Positive: Two distinct real roots
  • Zero: One repeated real root (parabola touches x-axis)
  • Negative: Two complex conjugate roots (parabola doesn't cross x-axis)

Examples

  • x² - 5x + 6 = 0 → x = 2 or x = 3 (discriminant = 1)
  • x² - 4x + 4 = 0 → x = 2 (discriminant = 0)
  • x² + x + 1 = 0 → x = -0.5 ± 0.866i (discriminant = -3)

Vertex Form

The vertex of the parabola y = ax² + bx + c is at (-b/2a, c - b²/4a). The axis of symmetry is x = -b/2a.

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Frequently Asked Questions

What is the quadratic formula?

The quadratic formula x = (-b ± √(b² - 4ac)) / 2a solves any quadratic equation ax² + bx + c = 0, where a ≠ 0.

What is the discriminant?

The discriminant is b² - 4ac. If positive, there are two real roots. If zero, one repeated root. If negative, two complex conjugate roots.

Can a equal zero?

No. If a = 0 the equation becomes bx + c = 0, which is linear, not quadratic.

What are complex roots?

When the discriminant is negative, the square root produces imaginary numbers. The roots are expressed as a ± bi, where i = √(-1).