A quadratic equation is a second-degree polynomial equation in a single variable with the general form:

\(ax^2 + bx + c = 0 \)

Here, x represents the variable, and a, b, and c are constants with \( a \ne 0 \) . The solutions to the quadratic equation, or the values of xx that satisfy the equation, can be found using the quadratic formula:

\( x = {-b \pm \sqrt{b^2-4ac} \over 2a} \)

The term inside the square root, \( b^2 − 4ac \), is called the discriminant. The discriminant determines the nature of the solutions:

If \( b^2 − 4ac > 0 \), the quadratic equation has two distinct real solutions.

If \( b^2 − 4ac = 0 \), the quadratic equation has exactly one real solution (a repeated or double root).

If \( b^2 − 4ac < 0 \), the quadratic equation has two complex conjugate solutions (no real solutions).

Quadratic equations frequently appear in various fields of mathematics and physics, and they describe a wide range of phenomena, including the motion of projectiles, the shape of certain curves, and the behavior of systems with quadratic relationships.