Quadratic Formula Calculator

A quadratic equation is a second-degree polynomial equation in a single variable x, with a ≠ 0. The standard form is ax² + bx + c = 0, where a, b, and c are coefficients. The graph of a quadratic equation is a parabola, which can open upward or downward depending on the sign of the coefficient a.

The quadratic formula is used to find the roots of a quadratic equation. It is expressed as x = (-b ± √(b² – 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0. The term under the square root, b² – 4ac, is called the discriminant.

The discriminant (Δ) is the expression b² – 4ac in the quadratic formula. It tells us about the nature of the roots: if Δ > 0, there are two distinct real roots; if Δ = 0, there is exactly one real root (a repeated root); and if Δ < 0, there are two complex conjugate roots.

The vertex of a parabola represented by the quadratic equation ax² + bx + c = 0 can be found using the formula h = -b / 2a for the x-coordinate. To find the y-coordinate (k), substitute the x-coordinate back into the equation: k = f(h). The vertex is the point (h, k) where the parabola changes direction.

Quadratic equations have numerous real-world applications, including calculating projectile motion (like the path of a thrown ball), determining areas and dimensions in architecture and engineering, optimizing profit and cost in business, analyzing electrical circuits, and modeling population growth. They are fundamental in physics, engineering, economics, and many other fields.