Quadratic Equations: Discriminant and Quadratic Formula

Quadratic Equations

Quadratic equations are mathematical expressions of the form ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable. These equations describe parabolas when graphed on the coordinate plane, with the vertex of the parabola located at (-b / 2a, f(-b / 2a)), where f() denotes the focus of the parabola. Quadratic equations have many applications in mathematics and science. Two important aspects of quadratic equations are the discriminant and the quadratic formula.

Discriminant

The discriminant of a quadratic equation is given by the formula b^2 - 4ac. It determines the nature of the roots of the equation. When the discriminant is non-negative (greater than or equal to 0), the equation has at least one real root. When the discriminant is 0, the equation has exactly one real root. When the discriminant is negative (less than 0), the equation has no real root.

Quadratic Formula

The quadratic formula is an equation that provides all possible values of x that make a quadratic equation true. It is derived from the quadratic formula, which is given by x = (-b ± sqrt(b^2 - 4ac)) / 2a. This formula allows us to find the roots of any quadratic equation by substituting the values of a, b, and c into the formula.

In summary, quadratic equations are important in mathematics and have various applications. The discriminant and the quadratic formula are crucial aspects of these equations, helping us understand their nature and find their roots.