Answer

Verified

447.6k+ views

**Hint:**Quadratic equations are the equation that contains at least one squared variable which is equal to zero. Quadratic equations are useful in our daily life; they are used to calculate areas, speed of the objects, projection, etc.

Quadratic equation is given as \[a{x^2} + bx + c = 0\]. This is the basic equation which contains a squared variable \[x\] and three constants a, b and c. The value of the \[x\] in the equation which makes the equation true is known as the roots of the equation. The numbers of roots in the quadratic equations are two as the highest power on the variable of the equation is x. The roots of the equation are given by the formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\], where \[{b^2} - 4ac\] tells the nature of the solution.

In the quadratic equation, \[a{x^2} + bx + c = 0\]the sum of the roots is given by \[ -

\dfrac{b}{a}\] whereas their products are given by \[\dfrac{c}{a}\].

In this question, it is already mentioned that one of the roots is the square of the other and so we need to carry out the calculation by taking only variable for the root of the equation $4{x^2} - 15x + 4p = 0$ and determining the relation between the roots and p.

**Complete step by step solution:**Let one of the roots of the equation $4{x^2} - 15x + 4p = 0$ be $m$ then, according to question the other root will be ${m^2}$.

Now, following the property of the quadratic equation that the product of the roots is equal to the ratio of the coefficient of ${x^0}$ and the coefficient of ${x^2}$.

Here, the coefficient of ${x^0}$is 4p and the coefficient of ${x^2}$ is 4.

Hence,

$

m \times {m^2} = \dfrac{{4p}}{4} \\

{m^3} = p - - - - (i) \\

$

Also, the sum of the roots of the quadratic equation is the negation of the ratio of the coefficient of $x$ and the coefficient of ${x^2}$.

Here, the coefficient of $x$is -15 and the coefficient of ${x^2}$ is 4.

Hence,

\[

m + {m^2} = - \left( {\dfrac{{ - 15}}{4}} \right) \\

{m^2} + m - \dfrac{{15}}{4} = 0 \\

4{m^2} + 4m - 15 = 0 \\

m = \dfrac{{ - 4 \pm \sqrt {{4^2} - 4(4)( - 15)} }}{{2(4)}} \\

= \dfrac{{ - 4 \pm \sqrt {16 + 240} }}{8} \\

= \dfrac{{ - 4 \pm 16}}{8} \\

= - \dfrac{5}{2},\dfrac{3}{2} - - - - (ii) \\

\]

By equation (i) and (ii) we get:

For $m = \dfrac{{ - 5}}{2}$; $p = {\left( {\dfrac{{ - 5}}{2}} \right)^3} = \dfrac{{ - 125}}{8}$

For $m = \dfrac{3}{2}$ ; $p = {\left( {\dfrac{3}{2}} \right)^3} = \dfrac{{27}}{8}$

Hence, the value of p can either be $\dfrac{{ - 125}}{8}$ or $\dfrac{{27}}{8}$.

Option C and D are correct.

**Note:**In the quadratic equation if \[{b^2} - 4ac > 0\]the equation will have two real roots. If it is equal \[{b^2} - 4ac = 0\] then the equation will have only one real root and when \[{b^2} - 4ac < 0\] then the root is in complex form.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Mark and label the given geoinformation on the outline class 11 social science CBSE

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

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

Trending doubts

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Which are the Top 10 Largest Countries of the World?

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

Give 10 examples for herbs , shrubs , climbers , creepers

Choose the antonym of the word given below Furious class 9 english CBSE

How do you graph the function fx 4x class 9 maths CBSE

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

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