Quadratic Equation Calculator

What is a Quadratic Equation Calculator?

A quadratic equation calculator is a tool that solves equations in the form ax² + bx + c = 0, where a, b, and c are coefficients and a ≠ 0. These equations are fundamental in algebra and appear in various real-world applications, from physics to engineering.

This calculator helps you find the roots (solutions) of quadratic equations using the quadratic formula. It provides both real and complex solutions, along with a step-by-step explanation of the calculation process.

Quadratic Equation Formulas

The quadratic formula is used to find the solutions of a quadratic equation:

x = (-b ± √(b² - 4ac)) / (2a)

Variables:

• x: The unknown variable (roots/solutions)
• a: Coefficient of x² (must not be zero)
• b: Coefficient of x
• c: Constant term
• Δ (Discriminant): b² - 4ac

The discriminant (Δ) determines the nature of the roots:

• If Δ > 0: Two distinct real roots
• If Δ = 0: One real root (repeated)
• If Δ < 0: Two complex conjugate roots

How to Calculate Quadratic Equations: Example

Let's solve the equation: 2x² + 5x - 3 = 0

1. Identify the coefficients:
   • a = 2
   • b = 5
   • c = -3
2. Calculate the discriminant:

   Δ = b² - 4ac Δ = 5² - 4(2)(-3) Δ = 25 + 24 Δ = 49

3. Apply the quadratic formula:

   x = (-5 ± √49) / (2 * 2) x = (-5 ± 7) / 4

4. Calculate both solutions:
   • x₁ = (-5 + 7) / 4 = 0.5
   • x₂ = (-5 - 7) / 4 = -3

Common Applications

• Physics: Calculating projectile motion, determining maximum height and range
• Engineering: Solving problems involving area, volume, and structural analysis
• Economics: Finding break-even points and profit maximization
• Architecture: Designing parabolic structures and calculating optimal dimensions
• Computer Graphics: Implementing collision detection and curve rendering