Answer

Verified

411.9k+ 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

The base of a right prism is a pentagon whose sides class 10 maths CBSE

A die is thrown Find the probability that the number class 10 maths CBSE

A mans age is six times the age of his son In six years class 10 maths CBSE

A started a business with Rs 21000 and is joined afterwards class 10 maths CBSE

Aasifbhai bought a refrigerator at Rs 10000 After some class 10 maths CBSE

Give a brief history of the mathematician Pythagoras class 10 maths CBSE

Trending doubts

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

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

Name 10 Living and Non living things class 9 biology CBSE

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

Write a letter to the principal requesting him to grant class 10 english CBSE

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

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