The Discriminant In Algebraic Equations: Importance And Applications

Discriminant

b²-4ac

In mathematics, the discriminant is a term used in algebraic equations, particularly in quadratic equations. The discriminant is denoted by the symbol Delta (Δ) and is found by taking the square root of the expression b2 – 4ac in a quadratic equation ax2 + bx + c = 0.

The discriminant plays an important role in determining the nature of the roots of a quadratic equation, whether they are real or imaginary, and whether they are equal or distinct. There are three possible cases:

1. If the discriminant is positive (Δ > 0), then the roots are real and distinct.
2. If the discriminant is zero (Δ = 0), then the roots are real and equal (also known as a double root or root of multiplicity 2).
3. If the discriminant is negative (Δ < 0), then the roots are complex conjugates (i.e. a + bi and a – bi, where a and b are real numbers and b is non-zero). The discriminant can also be used to determine the number of roots of a polynomial equation of degree n. For example, for a cubic equation (degree 3), if the discriminant is positive, then there are three real roots; if the discriminant is zero, then there are two real roots and one root of multiplicity 2; and if the discriminant is negative, then there is only one real root and two complex conjugate roots.

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