Quadratic Equation Solver

Solve quadratic equations of the form ax² + bx + c = 0. Get real and complex roots with step-by-step solutions.

Solve Quadratic Equation

Enter the coefficients for the quadratic equation:

ax² + bx + c = 0

The Quadratic Formula

The quadratic formula is used to solve quadratic equations of the form ax² + bx + c = 0:

x = [-b ± √(b² - 4ac)] / (2a)

The Discriminant

The expression inside the square root (b² - 4ac) is called the discriminant and tells us about the nature of the roots:

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

Example Solutions

Example 1: Two Real Roots

Solve: x² + 5x + 6 = 0

a = 1, b = 5, c = 6
Δ = 25 - 4×1×6 = 1
x = [-5 ± √1] / 2 = [-5 ± 1] / 2
x₁ = (-5 + 1) / 2 = -2
x₂ = (-5 - 1) / 2 = -3

Example 2: One Real Root

Solve: x² - 4x + 4 = 0

a = 1, b = -4, c = 4
Δ = 16 - 4×1×4 = 0
x = [4 ± √0] / 2 = 4/2 = 2

Example 3: Complex Roots

Solve: x² + 2x + 5 = 0

a = 1, b = 2, c = 5
Δ = 4 - 4×1×5 = -16
x = [-2 ± √(-16)] / 2 = [-2 ± 4i] / 2
x₁ = -1 + 2i
x₂ = -1 - 2i

Completing the Square

Another method to solve quadratic equations is completing the square:

ax² + bx + c = 0
x² + (b/a)x = -c/a
x² + (b/a)x + (b/(2a))² = -c/a + (b/(2a))²
(x + b/(2a))² = (b² - 4ac)/(4a²)
x + b/(2a) = ±√(b² - 4ac)/(2a)
x = [-b ± √(b² - 4ac)]/(2a)

Applications

Quadratic equations appear in many real-world applications:

  • 🏀 Physics: Projectile motion and trajectories
  • 💰 Finance: Compound interest calculations
  • 🏗️ Engineering: Structural analysis and optimization
  • 📊 Business: Break-even analysis and profit maximization

💡 Tip: Always check your solutions by substituting them back into the original equation to verify they work!