Question

How do you use the discriminant to determine the nature of the roots for $4{{x}^{2}}+15x+10=0$?

Answer 1

Hint: We start solving the problem by recalling the fact that the discriminant of the quadratic equation $a{{x}^{2}}+bx+c=0$ as $D={{b}^{2}}-4ac$. We then compare the given quadratic equation $4{{x}^{2}}+15x+10=0$ with $a{{x}^{2}}+bx+c=0$ to get the values of a, b and c. We then find the discriminant of the given quadratic equation using these values. We then compare this with the properties of discriminant to get to know about the properties of the roots of the given quadratic equation.

Complete step-by-step answer:
According to the problem, we are asked to find the discriminant of the quadratic equation $4{{x}^{2}}+15x+10=0$ and mention the properties of the roots of that quadratic equation.
We have given the quadratic equation $4{{x}^{2}}+15x+10=0$ ---(1).
We know that the discriminant of the quadratic equation $a{{x}^{2}}+bx+c=0$ is defined as $D={{b}^{2}}-4ac$. On comparing the given quadratic equation $4{{x}^{2}}+15x+10=0$ with $a{{x}^{2}}+bx+c=0$, we get $a=4$, $b=15$, $c=10$.
So, the discriminant of the given quadratic equation $4{{x}^{2}}+15x+10=0$ is $D={{15}^{2}}-4\left( 4 \right)\left( 10 \right)$.
$\Rightarrow D=225-160$.
$\Rightarrow D=65$, which is positively real and not a perfect square ---(1).
We know that the properties of the roots of the quadratic equation $a{{x}^{2}}+bx+c=0$ is defined as follow:
If $D < 0$, then the roots are complex and distinct.
If $D=0$, then the roots are rational and equal.
If $D > 0$ and perfect square, then the roots are distinct, real and rational.
If $D > 0$ and not a perfect square, then the roots are distinct, real and irrational.
From equation (1), we can see that $D > 0$ and not perfect squares. So, the roots are distinct real and irrational roots.
$\therefore $ The roots of the quadratic equation $4{{x}^{2}}+15x+10=0$ are distinct real and irrational roots.

Note: Whenever we get this type of problems, we first compare the given quadratic equation with the standard equation which leads us to the required answer. We should not confuse discriminant with $\sqrt{{{b}^{2}}-4ac}$ instead of ${{b}^{2}}-4ac$, which is the common mistake done by students. We should keep in mind to check whether the obtained discriminant is perfect square or not. Similarly, we can expect problems to check the properties of the roots of the quadratic equation ${{x}^{2}}+4x+4=0$.