Quadratic Solver
Solve quadratic equations
Enter Your Equation
ax² + bx + c = 0
Solution
Understanding Quadratic Equations
Quadratic equations show up more often than you'd think - calculating projectile motion, optimizing profits in business, even designing suspension bridges. The standard form is ax² + bx + c = 0, where a, b, c are numbers and a ≠ 0.
Our quadratic solver handles real and complex roots, shows the discriminant, and provides the vertex of the parabola. Perfect for students, engineers, and professionals.
How to Use This Quadratic Equation Solver
Step 1: Enter the coefficient 'a' (x² term - cannot be zero).
Step 2: Enter the coefficient 'b' (x term).
Step 3: Enter the constant 'c' (constant term).
Step 4: Click "Solve Equation" to see the roots.
Step 5: View roots, discriminant, vertex, and solution type.
Step 6: Use the Reset button to clear all inputs and start a new calculation.
Why Use a Quadratic Equation Solver?
✓ Instant Solutions
Get roots immediately. No manual calculations needed. Perfect for checking homework answers.
✓ Discriminant Analysis
Understand the nature of roots. See if solutions are real, repeated, or complex.
✓ Vertex Calculation
Find the vertex of the parabola. Know the minimum or maximum point of the quadratic.
✓ Complex Roots
Handle equations with imaginary solutions. Perfect for advanced math and engineering.
Discriminant Guide - What It Tells You
| Discriminant (Δ) | Nature of Roots | Graph Behavior |
|---|---|---|
| Δ > 0 | Two distinct real roots | Parabola crosses x-axis twice |
| Δ = 0 | One real root (double root) | Parabola touches x-axis at vertex |
| Δ < 0 | Two complex conjugate roots | Parabola never touches x-axis |
Real-World Applications of Quadratic Equations
- 🚀Projectile Motion: Calculating trajectory of thrown objects, rockets, and sports balls.
- 🚀Business & Finance: Profit optimization, break-even analysis, revenue maximization.
- 🚀Engineering: Bridge design, suspension cables, structural analysis, signal processing.
- 🚀Physics: Free-fall motion, kinetic energy, optics, electromagnetic waves.
Methods to Solve Quadratic Equations
1. Factoring
Best when equation factors easily. Find two numbers that multiply to ac and add to b.
2. Quadratic Formula
Works for ALL equations. x = [-b ± √(b²-4ac)] / 2a. Our calculator uses this method.
3. Completing the Square
Useful for deriving formula and for equations with perfect squares.
4. Graphing
Find x-intercepts of parabola. Good for visualizing solutions.
The Quadratic Formula
x = [-b ± √(b² - 4ac)] / (2a)
Example: x² - 3x + 2 = 0 → a=1, b=-3, c=2 → x = (3 ± √(9-8)) / 2 = (3 ± 1)/2 → x = 2 or x = 1