Quadratic Equation Solver
Calculation Results
Understanding the Quadratic Formula
A quadratic equation is a second-order polynomial equation in a single variable x, with the standard form:
ax² + bx + c = 0
where x represents an unknown, and a, b, and c represent known numbers, with a ≠ 0. If a = 0, the equation is linear, not quadratic.
How to Use This Algebra Calculator
To find the roots of your equation, simply follow these steps:
- Identify the coefficients a, b, and c from your equation. For example, in 2x² – 4x + 2 = 0, a=2, b=-4, and c=2.
- Enter the value of a into the first input field.
- Enter the value of b into the second input field.
- Enter the value of c into the third input field.
- Click the "Solve Equation" button to see the discriminant and the values of x.
The Math Behind the Solution
This calculator uses the Quadratic Formula to determine the value of x. The formula is derived from the process of completing the square and is universally valid for all quadratic equations:
x = (-b ± √(b² – 4ac)) / 2a
The term inside the square root, (b² – 4ac), is called the Discriminant. The value of the discriminant tells us what kind of roots the equation has:
- Positive Discriminant: The equation has two distinct real roots.
- Zero Discriminant: The equation has exactly one real root (a repeated root).
- Negative Discriminant: The equation has two complex roots (involving imaginary numbers).
Real-World Applications
Quadratic equations are not just abstract math problems; they describe many physical phenomena. Common applications include:
- Projectile Motion: Calculating the path of a ball thrown into the air or a missile fired.
- Area Calculations: Determining the dimensions of a plot of land or maximizing the area of a garden with a fixed perimeter.
- Physics and Engineering: Describing the behavior of springs, pendulums, and electronic circuits.