Quadratic Equation Solver

How to Use This Calculator

Enter the coefficients a, b, and c for your quadratic equation in the form ax² + bx + c = 0. Click "Calculate" to find the roots (solutions) along with step-by-step working. The calculator handles real and complex roots.

The Quadratic Formula

Understanding the Discriminant

The discriminant D = b² − 4ac determines the nature of the roots before you solve:

• D > 0: Two distinct real roots. The parabola crosses the x-axis at two points.
• D = 0: One repeated real root (a double root). The parabola just touches the x-axis at its vertex.
• D < 0: Two complex conjugate roots (no real x-intercepts). The parabola does not cross the x-axis.

Step-by-Step Worked Examples

Example 1: x² − 5x + 6 = 0 (a=1, b=−5, c=6)

1. D = (−5)² − 4(1)(6) = 25 − 24 = 1 (D > 0, two real roots)
2. x = (5 ± √1) / 2 = (5 ± 1) / 2
3. x₁ = 3, x₂ = 2

Example 2: x² + 2x + 1 = 0 (a=1, b=2, c=1)

1. D = 4 − 4 = 0 (double root)
2. x = −2 / 2 = −1

Example 3: x² + 1 = 0 (a=1, b=0, c=1)

1. D = 0 − 4 = −4 (D < 0, complex roots)
2. x = ±√(−4) / 2 = ±i

Real-World Applications

Quadratic equations appear in projectile motion (height = −½gt² + v₀t + h₀), area optimization problems (maximizing a rectangular enclosure for a given perimeter), revenue and profit calculations in economics (quadratic cost/revenue models), and engineering design (parabolic reflectors for satellite dishes and car headlights). The vertex of the parabola (h = −b/2a, k = f(h)) gives the maximum or minimum point, which is the key to optimization problems.