Quadratic Equation Solver
Solve quadratic equations of the form ax² + bx + c = 0
Solution
Enter coefficients to solve the quadratic equation.
Related Math Tools
How to Use the Quadratic Equation Solver
- Enter coefficients: Input values for a, b, and c in the standard form ax² + bx + c = 0
- Coefficient a: Must not be zero (otherwise it's not a quadratic equation)
- Get solutions: The calculator uses the quadratic formula: x = (-b ± √(b²-4ac)) / 2a
- Understand results: Solutions can be real numbers or complex numbers depending on the discriminant
Example: For x² - 5x + 6 = 0, enter a=1, b=-5, c=6. Solutions: x₁ = 3, x₂ = 2.
Understanding Quadratic Equations
A quadratic equation is a polynomial equation of degree 2, written in the standard form ax² + bx + c = 0, where a ≠ 0. The solutions (roots) of a quadratic equation can be found using the quadratic formula, factoring, or completing the square.
The Quadratic Formula
x = (-b ± √(b² - 4ac)) / 2a
The Discriminant
The discriminant (b² - 4ac) determines the nature of the roots:
- Positive discriminant: Two distinct real roots
- Zero discriminant: One repeated real root
- Negative discriminant: Two complex conjugate roots
Real-World Applications
- Physics: Projectile motion, free fall calculations
- Engineering: Optimization problems, structural design
- Finance: Profit maximization, break-even analysis
- Geometry: Area and volume optimization problems
Step-by-Step Example
Solve: 2x² - 7x + 3 = 0
Step 1: Identify coefficients: a = 2, b = -7, c = 3
Step 2: Calculate discriminant: b² - 4ac = (-7)² - 4(2)(3) = 49 - 24 = 25
Step 3: Apply quadratic formula: x = (7 ± √25) / 4 = (7 ± 5) / 4
Step 4: Find roots: x₁ = (7 + 5) / 4 = 3, x₂ = (7 - 5) / 4 = 0.5