Answer

Verified

411k+ views

**Hint:**We start solving the problem by using the fact that discriminant should be greater than zero for roots to become real and distinct. We now assume a positive number for discriminant and substitute this in the roots of the quadratic equation. We get two cases related to the denominator of the roots to find the characteristics of the roots of the equation.

**Complete step by step answer:**

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.

**So, the correct answer is “Option C”.**

**Note:**We should not take the value of discriminant greater than or equal to zero here $\left( D\ge 0 \right)$ for this problem as the problem clearly stated that the roots are real and distinct. From this problem we have seen that both roots will not be greater than or less than the value of $\dfrac{-b}{2a}$. Similarly we can expect problems to find about properties of roots by giving about the properties of the ‘b’.

Recently Updated Pages

What are the Advantages and Disadvantages of Algorithm

How do you write 0125 in scientific notation class 0 maths CBSE

The marks obtained by 50 students of class 10 out of class 11 maths CBSE

You are awaiting your class 10th results Meanwhile class 7 english CBSE

Which one of the following was not the cause of the class 10 social science CBSE

Which one of the following cities is not located on class 10 social science CBSE

Trending doubts

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Select the word that is correctly spelled a Twelveth class 10 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Write the 6 fundamental rights of India and explain in detail

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE