Quadratic Formula Calculator

Solve quadratic equations using the quadratic formula. Essential for algebra, physics, engineering, and mathematical analysis.

How to Use Quadratic Formula Calculator

1. Enter Coefficient a: Input the coefficient of x² term
2. Enter Coefficient b: Input the coefficient of x term
3. Enter Coefficient c: Input the constant term
4. Click "Solve": Get instant roots and discriminant
5. View Results: See roots, discriminant, and nature of roots

Features

- Quadratic Formula: Solve ax² + bx + c = 0
- Discriminant Analysis: Determine nature of roots
- Complex Roots: Handle imaginary solutions
- Step-by-Step: See calculation process
- Educational: Learn quadratic equation concepts

Quadratic Formula

Standard Form
[ ax^2 + bx + c = 0 ]

Quadratic Formula
[ x = rac{-b pm sqrt{b^2 - 4ac}}{2a} ]

Discriminant
[ Delta = b^2 - 4ac ]

Nature of Roots

Positive Discriminant (Δ > 0)
- Two real and distinct roots
- Parabola crosses x-axis twice
- Example: x² - 5x + 6 = 0 has roots x = 2, 3

Zero Discriminant (Δ = 0)
- One real root (repeated)
- Parabola touches x-axis once
- Example: x² - 4x + 4 = 0 has root x = 2

Negative Discriminant (Δ < 0)
- Two complex conjugate roots
- Parabola doesn't cross x-axis
- Example: x² + 4 = 0 has roots x = ±2i

Practical Applications

Physics
- Projectile Motion: Height vs. time equations
- Optics: Lens and mirror equations
- Mechanics: Energy and motion problems
- Electrical Circuits: Resonance frequency calculations

Engineering
- Structural Analysis: Beam deflection equations
- Control Systems: Stability analysis
- Signal Processing: Filter design
- Optimization: Finding maximum/minimum values

Economics and Finance
- Profit Optimization: Revenue and cost functions
- Market Equilibrium: Supply and demand curves
- Investment Analysis: Return calculations
- Risk Assessment: Probability distributions

Quadratic Examples

Example 1: Real Roots
- Equation: x² - 5x + 6 = 0
- Coefficients: a=1, b=-5, c=6
- Discriminant: 25 - 24 = 1
- Roots: x = 2, x = 3

Example 2: Repeated Root
- Equation: x² - 4x + 4 = 0
- Coefficients: a=1, b=-4, c=4
- Discriminant: 16 - 16 = 0
- Root: x = 2 (repeated)

Example 3: Complex Roots
- Equation: x² + 4x + 5 = 0
- Coefficients: a=1, b=4, c=5
- Discriminant: 16 - 20 = -4
- Roots: x = -2 ± i

Solving Methods

Factoring Method
When possible, factor the quadratic:
- x² - 5x + 6 = (x - 2)(x - 3) = 0
- Roots: x = 2, x = 3

Completing the Square
Transform to perfect square form:
- x² + bx + c = (x + b/2)² - (b/2)² + c

Graphical Method
Find x-intercepts of parabola y = ax² + bx + c

Advanced Concepts

Vertex Form
[ y = a(x - h)^2 + k ]
Where (h, k) is the vertex of the parabola

Sum and Product of Roots
- Sum: x₁ + x₂ = -b/a
- Product: x₁ × x₂ = c/a

Axis of Symmetry
[ x = -rac{b}{2a} ]

Tips for Quadratic Equations

Common Mistakes to Avoid
- Zero Coefficient: 'a' cannot be zero
- Sign Errors: Watch negative signs carefully
- Discriminant: Calculate b² - 4ac correctly
- Division: Remember to divide by 2a

Problem-Solving Strategies
- Identify Type: Recognize quadratic form
- Choose Method: Select appropriate solving method
- Check Solutions: Verify roots in original equation
- Interpret Results: Understand meaning of solutions

Conclusion

Quadratic equations are fundamental to mathematics and its applications. Mastering the quadratic formula enables you to solve a wide range of problems in science, engineering, and economics. This calculator provides essential tools for quadratic analysis, helping you work confidently with quadratic equations in academic and professional contexts.