Solving Quadratic Equations
The quadratic formula works for any ax² + bx + c = 0. The discriminant b² - 4ac determines the nature of roots.
Methods to Solve
| Method | Best For |
|---|---|
| Factoring | Simple integer roots |
| Quadratic Formula | Any quadratic |
| Completing the Square | Deriving vertex form |
How to Use This Quadratic Equation Solver
Enter the three coefficients of your quadratic equation (ax² + bx + c = 0). The solver provides exact roots, a step-by-step solution, and graphs the parabola.
Formula & How It Works
Step 1: Calculate discriminant Δ = b² – 4ac. Step 2: If Δ ≥ 0, roots = (-b ± √Δ) / 2a. If Δ < 0, roots are complex: (-b ± i√|Δ|) / 2a.
Calculation Example
2x² + 3x – 5 = 0: Δ = 9+40 = 49. x = (-3 ± 7)/4 → x = 1 or x = -2.5. The parabola opens upward (a > 0) and crosses the x-axis at these points.
Expert Tips
Quadratics model projectile motion (physics), profit optimization (business), and curve fitting (data science). The vertex formula x = -b/2a gives the axis of symmetry.
Frequently Asked Questions
What is the quadratic formula?
x = (-b ± √(b²-4ac)) / 2a. It finds the roots of any quadratic equation ax² + bx + c = 0.
What does the discriminant tell us?
D > 0: two real roots. D = 0: one double root. D < 0: two complex roots.
What is the vertex?
The vertex is the maximum or minimum point of the parabola at x = -b/(2a).