Question

Find the characteristics of the roots of the quadratic equation $3{{x}^{2}}-7x+2$?

Answer 1

Hint: We first describe the use of discriminant in the polynomials. Then we find the discriminant for the quadratic equation $3{{x}^{2}}-7x+2$. We explain the conditions for equal/unequal, rational/irrational roots in that quadratic equation.

Complete step by step solution:
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 $D={{b}^{2}}-4ac$.
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.
For $3{{x}^{2}}-7x+2$, $D={{\left( -7 \right)}^{2}}-4\times 3\times 2=25$ is a perfect square. Therefore, the roots are rational but unequal.

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.