Mastering The Quadratic Equation: The Importance Of Discriminant In Determining The Roots And Nature Of Solutions

Discriminant

b²-4ac

In mathematics, a discriminant is a formula that is used to evaluate the nature of roots of a quadratic equation. A quadratic equation is an equation of the form ax²+bx+c=0, where a, b, and c are constants. The discriminant of the quadratic equation is defined as b²-4ac.

The discriminant is used to determine the number and nature of the roots of a quadratic equation. There are three cases:

1. If the discriminant is positive, i.e., b²-4ac > 0, then the quadratic equation has two distinct real roots.

2. If the discriminant is zero, i.e., b²-4ac = 0, then the quadratic equation has one real root that is repeated, called a double root.

3. If the discriminant is negative, i.e. b²-4ac < 0, then the quadratic equation has two imaginary roots, which are complex conjugates of each other. The discriminant plays a crucial role not only in solving quadratic equations, but also in many areas of mathematics, such as geometry, number theory, and calculus. It is used to determine the nature of critical points in optimization problems, to find the nature of conic sections and to solve problems involving quadratic Diophantine equations.

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