Quadratic Formula Calculator
Solve quadratic equations in the form ax² + bx + c = 0 and analyze their properties with our comprehensive calculator.
Understanding Quadratic Equations
Our Quadratic Formula Calculator helps you solve any quadratic equation in the form ax² + bx + c = 0 and understand its properties. The calculator uses the quadratic formula to find roots and provides additional information about the equation.
The Quadratic Formula
For any quadratic equation ax² + bx + c = 0, the solutions are given by:
x = (-b ± √(b² - 4ac)) / (2a)
The term under the square root (b² - 4ac) is called the discriminant (Δ) and determines the nature of the roots:
- If Δ > 0: Two distinct real roots
- If Δ = 0: One real root (double root)
- If Δ < 0: Two complex conjugate roots
Key Concepts
Vertex Form:
A quadratic function can be written in vertex form:
y = a(x - h)² + k
where (h, k) is the vertex of the parabola.
Axis of Symmetry:
The vertical line passing through the vertex:
x = -b/(2a)
The parabola is symmetric about this line.
How to Use the Calculator:
- Enter the coefficient a (the coefficient of x²)
- Enter the coefficient b (the coefficient of x)
- Enter the coefficient c (the constant term)
- Select the desired number of decimal places for the results
- Click "Solve Quadratic Equation" to see the complete solution
Applications of Quadratic Equations
Quadratic equations are used in many real-world applications:
- Physics: projectile motion, falling objects
- Engineering: structural designs, electrical circuits
- Economics: profit maximization, supply and demand models
- Finance: interest calculations, break-even analysis
- Geometry: area calculations, distance problems
Common Examples
Example 1: x² + 5x + 6 = 0
Here, a=1, b=5, c=6
Discriminant = 5² - 4(1)(6) = 25 - 24 = 1
Solutions: x = -2 and x = -3
Example 2: 2x² - 4x + 2 = 0
Here, a=2, b=-4, c=2
Discriminant = (-4)² - 4(2)(2) = 16 - 16 = 0
Solution: x = 1 (double root)