How to Use the Quadratic Formula Calculator

Enter the coefficients a, b, and c from your quadratic equation in standard form (ax² + bx + c = 0). The calculator will find the roots using the quadratic formula x = (-b ± √(b² - 4ac)) / 2a and provide additional insights about the parabola.

Understanding the Discriminant

The discriminant Δ = b² - 4ac determines the nature of the roots:

Δ > 0: Two distinct real roots
Δ = 0: One repeated real root (vertex touches x-axis)
Δ < 0: Two complex conjugate roots

Parabola Properties

Every quadratic equation represents a parabola when graphed. The calculator provides:

Vertex: The highest or lowest point of the parabola at (-b/2a, f(-b/2a))
Axis of Symmetry: The vertical line x = -b/2a through the vertex
Y-Intercept: Where the parabola crosses the y-axis (0, c)
Direction: Opens upward if a > 0, downward if a < 0

Vieta's Formulas

For a quadratic equation ax² + bx + c = 0 with roots x₁ and x₂:

Sum of roots: x₁ + x₂ = -b/a
Product of roots: x₁ × x₂ = c/a

These relationships hold true even when the roots are complex numbers.

Common Applications

Quadratic equations appear in many real-world scenarios:

Projectile motion and ballistics
Area and geometry optimization problems
Profit and revenue calculations in business
Signal processing and electronics
Physics problems involving acceleration