Question

If \[{\rm{\alpha ,\beta }}\] are the roots of the quadratic equation \[{{\rm{x}}^2}{\rm{ + px + q = 0}}\], then the values of \[{{\rm{\alpha }}^3}{\rm{ + }}{{\rm{\beta }}^3}\] and \[{{\rm{\alpha }}^4}{\rm{ + }}{{\rm{\alpha }}^2}{{\rm{\beta }}^2}{\rm{ + }}{{\rm{\beta }}^4}\] are respectively.
A) \[3{\rm{pq}} - {{\rm{p}}^3}{\rm{and }}{{\rm{p}}^4} - 3{{\rm{p}}^2}{\rm{q}} + 3{{\rm{q}}^2}\]
B) \[ - {\rm{p}}(3{\rm{q}} - {{\rm{p}}^2}){\rm{and }}({{\rm{p}}^2} - {\rm{q}})({{\rm{p}}^2} + 3{\rm{q}})\]
C) \[{\rm{pq}} - 4{\rm{and }}{{\rm{p}}^4} - {{\rm{q}}^4}\]
D) \[3{\rm{pq}} - {{\rm{p}}^3}{\rm{and }}({{\rm{p}}^2} - {\rm{q}})({{\rm{p}}^2} - 3{\rm{q}})\]

Answer 1

Hint:
Here, we have to use the basic concept of the roots of the quadratic equation. Then with the concept of the roots of the quadratic equation, we will get two basic equations showing the relation between \[{\rm{\alpha ,\beta ,p and q}}\]. So, by simply using that relation in the equations we will be able to find out the value of the given equations.

Complete step by step solution:
It is given that \[{\rm{\alpha ,\beta }}\] are the roots of the quadratic equation \[{{\rm{x}}^2}{\rm{ + px + q = 0}}\]
We know that the sum of the roots of a quadratic equation is equal to the value of the coefficient of \[{\rm{x}}\] and the product of the roots of the quadratic equation is equal to the value of the constant term in the quadratic equation. So, we will get \[{\rm{\alpha + \beta = }} - {\rm{p and \alpha \beta = q}}\].
So now we have to use this relation to find out the value of the given functions. Therefore, taking the first equation \[{{\rm{\alpha }}^3}{\rm{ + }}{{\rm{\beta }}^3}\]
We can write the above equation as
\[ \Rightarrow {\rm{(}}{{\rm{\alpha }}^3}{\rm{ + }}{{\rm{\beta }}^3}) = {({\rm{\alpha + \beta }})^3} - 3{\rm{\alpha \beta }}({\rm{\alpha + \beta }})\]
Now putting the values of the \[{\rm{\alpha + \beta = }} - {\rm{p and \alpha \beta = q}}\] in the above equation, we get
\[ \Rightarrow {\rm{(}}{{\rm{\alpha }}^3}{\rm{ + }}{{\rm{\beta }}^3}) = {( - {\rm{p}})^3} - 3{\rm{q}}( - {\rm{p}}) = - {{\rm{p}}^3} + 3{\rm{pq = }}3{\rm{pq}} - {{\rm{p}}^3}\]
\[ \Rightarrow {\rm{(}}{{\rm{\alpha }}^3}{\rm{ + }}{{\rm{\beta }}^3}){\rm{ = }}3{\rm{pq}} - {{\rm{p}}^3}\]
Now for the second equation and we have to simplify that equation, we get
\[ \Rightarrow {{\rm{\alpha }}^4}{\rm{ + }}{{\rm{\alpha }}^2}{{\rm{\beta }}^2}{\rm{ + }}{{\rm{\beta }}^4} = {{\rm{\alpha }}^4}{\rm{ + 2}}{{\rm{\alpha }}^2}{{\rm{\beta }}^2}{\rm{ + }}{{\rm{\beta }}^4} - {{\rm{\alpha }}^2}{{\rm{\beta }}^2}{\rm{ = (}}{{\rm{\alpha }}^2}{\rm{ + }}{{\rm{\beta }}^2}{{\rm{)}}^2} - {({\rm{\alpha \beta )}}^2}\]
\[ \Rightarrow {{\rm{\alpha }}^4}{\rm{ + }}{{\rm{\alpha }}^2}{{\rm{\beta }}^2}{\rm{ + }}{{\rm{\beta }}^4} = {{\rm{(}}{{\rm{\alpha }}^2}{\rm{ + }}{{\rm{\beta }}^2}{\rm{ + 2\alpha \beta }} - 2{\rm{\alpha \beta )}}^2} - {({\rm{\alpha \beta )}}^2} = {\left[ {{{({\rm{\alpha + \beta }})}^2}2{\rm{\alpha \beta }}} \right]^2} - {({\rm{\alpha \beta )}}^2}\]
Now putting the values of the \[{\rm{\alpha + \beta = }} - {\rm{p and \alpha \beta = q}}\] in the above equation, we get
\[ \Rightarrow {{\rm{\alpha }}^4}{\rm{ + }}{{\rm{\alpha }}^2}{{\rm{\beta }}^2}{\rm{ + }}{{\rm{\beta }}^4} = {\left[ {{{( - {\rm{p}})}^2} - 2{\rm{q}}} \right]^2} - {({\rm{q)}}^2} = {({{\rm{p}}^2} - 2{\rm{q}})^2} - {{\rm{q}}^2}\]
Now by simply expanding the square of the bracket term, we get
\[ \Rightarrow {{\rm{\alpha }}^4}{\rm{ + }}{{\rm{\alpha }}^2}{{\rm{\beta }}^2}{\rm{ + }}{{\rm{\beta }}^4} = {{\rm{p}}^4} - 4{{\rm{p}}^2}{\rm{q + 4}}{{\rm{q}}^2} - {{\rm{q}}^2} = {{\rm{p}}^4} - 4{{\rm{p}}^2}{\rm{q + 3}}{{\rm{q}}^2}\]
\[ \Rightarrow {{\rm{\alpha }}^4}{\rm{ + }}{{\rm{\alpha }}^2}{{\rm{\beta }}^2}{\rm{ + }}{{\rm{\beta }}^4} = {{\rm{p}}^4} - 3{{\rm{p}}^2}{\rm{q}} - {{\rm{p}}^2}{\rm{q + 3}}{{\rm{q}}^2} = {{\rm{p}}^2}({{\rm{p}}^2} - 3{\rm{q}}) - {\rm{q}}({{\rm{p}}^2} - 3{\rm{q}}) = ({{\rm{p}}^2} - {\rm{q}})({{\rm{p}}^2} - 3{\rm{q}})\]
\[ \Rightarrow {{\rm{\alpha }}^4}{\rm{ + }}{{\rm{\alpha }}^2}{{\rm{\beta }}^2}{\rm{ + }}{{\rm{\beta }}^4} = ({{\rm{p}}^2} - {\rm{q}})({{\rm{p}}^2} - 3{\rm{q}})\]
Hence the values of the given equations are
\[ \Rightarrow {\rm{(}}{{\rm{\alpha }}^3}{\rm{ + }}{{\rm{\beta }}^3}){\rm{ = }}3{\rm{pq}} - {{\rm{p}}^3}\]
\[ \Rightarrow {{\rm{\alpha }}^4}{\rm{ + }}{{\rm{\alpha }}^2}{{\rm{\beta }}^2}{\rm{ + }}{{\rm{\beta }}^4} = ({{\rm{p}}^2} - {\rm{q}})({{\rm{p}}^2} - 3{\rm{q}})\]

So, option D is the correct option.

Note:
Quadratic equation is an equation in which the highest exponent of the variable x is two and a quadratic equation has only two roots. Roots are those values of the equation where the value of the equation becomes zero. For any equation numbers of roots are always equal to the value of the highest exponent of the variable x.
We should know that algebraic identities in maths refer to an equation that is always true regardless of the values assigned to the variables. Algebraic identities are equations where the value of the left-hand side of the equation is identically equal to the value of the right-hand side of the equation. An algebraic identity is an equality that holds for any values of its variables.