Question

How can you tell when the roots are equal/unequal, rational/irrational and how many there are from the discriminant?

Answer 1

Hint: We first describe the use of discriminant in the polynomials. Then we find the discriminant for quadratic and cubic equations. We explain the conditions for equal/unequal, rational/irrational roots in case of quadratic equations.

Complete step by step answer:
Discriminant, in mathematics, a parameter of an object or system calculated as an aid to its classification or solution. In the case of a quadratic equation $a{{x}^{2}}+bx+c=0$ the discriminant is $D={{b}^{2}}-4ac$; for a cubic equation $a{{x}^{3}}+b{{x}^{2}}+cx+d=0$, the discriminant is $D=18abcd-4{{b}^{3}}d+{{b}^{2}}{{c}^{2}}-4a{{c}^{3}}-27{{a}^{2}}{{d}^{2}}$.
We will mainly discuss the quadratic equation, its discriminant and how characteristics of the roots are related to the discriminant.
We know for a general equation of quadratic $a{{x}^{2}}+bx+c=0$, the value of the roots of x will be $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$.
The roots of a quadratic equation with real coefficients are real and distinct if $D={{b}^{2}}-4ac>0$.
Roots are real but equal if $D={{b}^{2}}-4ac=0$
Roots are a conjugate pair of complex roots if $D={{b}^{2}}-4ac<0$.
The roots will be rational when $D={{b}^{2}}-4ac$ is a perfect square. If the discriminant is not a perfect square then the roots are irrational.
The number of roots is not dependent on the discriminant as that depends on the power of the equation. If the power of the equation is n then it has n roots. It can be both real and imaginary.

Note: The roots are equal when $D={{b}^{2}}-4ac=0$. Although the roots are the same, the number of roots will always be two. For our convenience we don’t use the root values twice but we can’t say that the number of roots for that quadratic equation is one as the roots are equal.