Answer

Verified

428.7k+ views

**Hint:**In this particular question use the concept that in a quadratic equation the sum of the roots is the ratio of the negative times the coefficient of x to the coefficient of ${x^2}$ and the product of the roots is the ratio of the constant term to the coefficient of ${x^2}$, so use these concepts to reach the solution of the question.

__Complete step-by-step answer__:Given data:

The sum and product of roots of a quadratic equation are $ - \dfrac{7}{2}$ and $\dfrac{5}{2}$ respectively.

Let the quadratic equation be $a{x^2} + bx + c = 0$, where and a, b, and c are the constant real parameters.

Let the roots of this quadratic equation be P and Q.

Now as we know that in a quadratic equation the sum of the roots is the ratio of the negative times the coefficient of x to the coefficient of ${x^2}$.

So the sum of the roots is,

$ \Rightarrow P + Q = \dfrac{{{\text{ - coefficient of }}x}}{{{\text{coefficient of }}{x^2}}}$

In the above quadratic equation the coefficient of x is b and the coefficient of ${x^2}$ is a.

Therefore, P + Q = $\dfrac{{ - b}}{a}$

Now it is given that the sum of the roots is $ - \dfrac{7}{2}$

Therefore, P + Q = $ - \dfrac{7}{2}$

$ \Rightarrow \dfrac{{ - b}}{a} = - \dfrac{7}{2}$

$ \Rightarrow \dfrac{b}{a} = \dfrac{7}{2}$……………. (1)

Now as we know that in a quadratic equation the product of the roots is the ratio of the constant term to the coefficient of ${x^2}$.

So the product of the roots is,

$ \Rightarrow PQ = \dfrac{{{\text{constant term}}}}{{{\text{coefficient of }}{x^2}}}$

Therefore, PQ = $\dfrac{c}{a}$

Now it is given that the product of the roots is $\dfrac{5}{2}$

$ \Rightarrow PQ = \dfrac{5}{2}$

$ \Rightarrow \dfrac{c}{a} = \dfrac{5}{2}$……………….. (2)

Now divide the quadratic equation by a throughout we have,

$ \Rightarrow {x^2} + \dfrac{b}{a}x + \dfrac{c}{a} = 0$

Now substitutes the value of $\dfrac{b}{a}{\text{ and }}\dfrac{c}{a}$ from equation (1) and (2) in the above equation we have,

$ \Rightarrow {x^2} + \dfrac{7}{2}x + \dfrac{5}{2} = 0$

Now multiply by 2 throughout we have,

$ \Rightarrow 2{x^2} + 7x + 5 = 0$

So this is the required quadratic equation.

**Hence option (a) is the correct answer.**

**Note**:We can also solve this problem directly, if the sum and the product of the roots of the quadratic equation are given then we directly write the quadratic equation which is given as, ${x^2} + \left( { - {\text{sum of the roots}}} \right)x + \left( {{\text{product of the roots}}} \right) = 0$ so simply substitute the given values in this equation and simplify we will get the required quadratic equation.

Recently Updated Pages

What number is 20 of 400 class 8 maths CBSE

Which one of the following numbers is completely divisible class 8 maths CBSE

What number is 78 of 50 A 32 B 35 C 36 D 39 E 41 class 8 maths CBSE

How many integers are there between 10 and 2 and how class 8 maths CBSE

The 3 is what percent of 12 class 8 maths CBSE

Find the circumference of the circle having radius class 8 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

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

One cusec is equal to how many liters class 8 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

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

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

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