Quadratic Formula Calculator

Solve quadratic equations using the quadratic formula. Calculate roots, discriminant, vertex, and axis of symmetry with step-by-step solutions.

Calculator

Cannot be zero

Enter the coefficients a, b, and c of the quadratic equation ax² + bx + c = 0 to solve for x.

Complete User Guide

What is the Quadratic Formula?

The quadratic formula is used to solve quadratic equations of the form ax² + bx + c = 0, where a, b, and c are coefficients and a ≠ 0.

x = (-b ± √(b² - 4ac)) / (2a)

The Discriminant

The discriminant (Δ = b² - 4ac) determines the nature of the solutions:

  • Δ > 0: Two distinct real solutions
  • Δ = 0: One real solution (repeated root)
  • Δ < 0: Two complex solutions

Additional Information

  • Vertex: The highest or lowest point of the parabola
  • Axis of Symmetry: The vertical line that divides the parabola into two equal halves
  • Parabola: The graph of a quadratic equation is a parabola

Examples

Example 1: Two Real Solutions

Equation: x² - 5x + 6 = 0

Coefficients: a = 1, b = -5, c = 6

Discriminant: Δ = 1 > 0

Solutions: x = 2, x = 3

Example 2: One Real Solution

Equation: x² - 4x + 4 = 0

Coefficients: a = 1, b = -4, c = 4

Discriminant: Δ = 0

Solution: x = 2 (repeated root)

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