Quadratic Formula Calculator

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

Worked Example

Solve x² - 5x + 6 = 0. Here a = 1, b = -5, c = 6, so the discriminant is Δ = (-5)² - 4(1)(6) = 25 - 24 = 1. The formula gives x = (5 ± √1) / 2, so x₁ = 3 and x₂ = 2. Check with Vieta's formulas: 3 + 2 = 5 = -b/a and 3 × 2 = 6 = c/a. The parabola opens upward with vertex at (2.5, -0.25).