Quadratic Equation Solver

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree 2, meaning the highest exponent of the variable (usually x) is 2. The standard form is:

$ax^2 + bx + c = 0$

Where:

• a, b, and c are coefficients (real numbers), and a ≠ 0.
• x is the unknown variable.

The Quadratic Formula

The most universal method to solve any quadratic equation is using the Quadratic Formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

This formula provides the solutions (roots) for any quadratic equation, whether the roots are real or complex.

The Discriminant

The term under the square root, $b^2 - 4ac$, is called the discriminant ($\Delta$). It tells us about the nature of the roots:

• If $\Delta > 0$, there are two distinct real roots.
• If $\Delta = 0$, there is exactly one real root (a repeated root).
• If $\Delta < 0$, there are no real roots (the roots are complex conjugates).

Applications

Quadratic equations appear frequently in physics, engineering, and economics. Examples include:

• Projectile Motion: Calculating the trajectory of a thrown object.
• Area Problems: Finding dimensions of shapes with a given area.
• Profit Maximization: Determining the production level that yields maximum profit.