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(a) Both roots are greater than $-\dfrac{b}{2a}$.

(b) Both roots are less than $-\dfrac{b}{2a}$.

(c) One of the roots exceeds $-\dfrac{b}{2a}$.

(d) None of the above.

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Given that we have a quadratic equation $a{{x}^{2}}+bx+c=0$ and it has real and distinct roots. We need to find characteristics of both roots.

We know that discriminant should be greater than zero for a quadratic equation to have real and distinct roots.

We know that discriminant $D=\sqrt{{{b}^{2}}-4ac}$.

We have got $\sqrt{{{b}^{2}}-4ac}>0$---(1).

We know that roots of quadratic equation $a{{x}^{2}}+bx+c=0$ are $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$.

We have the roots of the quadratic equation $a{{x}^{2}}+bx+c=0$ as $\dfrac{-b}{2a}\pm \dfrac{\sqrt{{{b}^{2}}-4ac}}{2a}$.

Let us assume that the value of the discriminant be ‘p’$\left( p>0 \right)$. So, we get the roots of the quadratic equation $a{{x}^{2}}+bx+c=0$ as $\dfrac{-b}{2a}\pm \dfrac{p}{2a}$.

We need to solve for two cases of ‘a’. (i) ‘a’ is positive (ii) ‘a’ is negative.

Let us assume the value of ‘a’ is positive. This makes $\dfrac{p}{2a}$ also a positive number.

We have the roots of the quadratic equation $a{{x}^{2}}+bx+c=0$ as $\dfrac{-b}{2a}\pm \dfrac{p}{2a}$. If $\dfrac{p}{2a}$ is added to $\dfrac{-b}{2a}$, then the value of the root exceeds $\dfrac{-b}{2a}$. If $\dfrac{p}{2a}$ is subtracted to $\dfrac{-b}{2a}$, then the value of the root will be less than $\dfrac{-b}{2a}$.

Let us assume the value of ‘a’ is negative. This makes $\dfrac{p}{2a}$ also a negative number

We have the roots of the quadratic equation $a{{x}^{2}}+bx+c=0$ as $\dfrac{-b}{2a}\pm \dfrac{p}{2a}$. If $\dfrac{p}{2a}$ is subtracted to $\dfrac{-b}{2a}$, then the value of the root exceeds $\dfrac{-b}{2a}$. If $\dfrac{p}{2a}$ is added to $\dfrac{-b}{2a}$, then the value of the root will be less than $\dfrac{-b}{2a}$.

We can see that from both cases of ‘a’, we got the value of one of the roots exceeds $\dfrac{-b}{2a}$ and other root is less than $\dfrac{-b}{2a}$.

∴ We get one of the roots exceed $\dfrac{-b}{2a}$ and another root is not.