If $\alpha ,\beta $ be the roots of ${x^2} - px + q = 0$and $\alpha ',\beta '$ are the roots of ${x^2} - p'x + q' = 0$, find the value of${\left( {\alpha - \alpha '} \right)^2} + {\left( {\beta - \alpha '} \right)^2} + {\left( {\alpha - \beta '} \right)^2} + {\left( {\beta - \beta '} \right)^2}$
Last updated date: 23rd Mar 2023
•
Total views: 307.2k
•
Views today: 2.85k
Answer
307.2k+ views
Hint: Use the properties of sum of the roots and product of the roots of the given quadratic polynomial equation.
Given $\alpha ,\beta $ are the roots of the quadratic equation ${x^2} - px + q = 0$and $\alpha ',\beta '$are the roots of the quadratic equation${x^2} - p'x + q' = 0$.
So, for the quadratic equation ${x^2} - px + q = 0$
Sum of the roots is $\alpha + \beta = p$ and product of the roots is $\alpha \beta = q$
Similarly, for the quadratic equation ${x^2} - p'x + q' = 0$
Sum of the roots is \[\alpha ' + \beta ' = p'\] and product of the roots is $\alpha '\beta ' = q'$
Now consider ${\left( {\alpha - \alpha '} \right)^2} + {\left( {\beta - \alpha '} \right)^2} + {\left( {\alpha - \beta '} \right)^2} + {\left( {\beta - \beta '} \right)^2}$
\[ = {\alpha ^2} + \alpha {'^2} - 2\alpha \alpha ' + {\beta ^2} + \alpha {'^2} - 2\beta \alpha ' + {\alpha ^2} + \beta {'^2} - 2\alpha \beta ' + {\beta ^2} + \beta {'^2} - 2\beta \beta '\]
Grouping the terms, we have
$ = 2\left( {{\alpha ^2} + \alpha {'^2} + {\beta ^2} + \beta {'^2}} \right) - 2\left( {\alpha + \beta } \right)\left( {\alpha ' + \beta '} \right)$
$ = 2\left[ {\left\{ {{{\left( {\alpha + \beta } \right)}^2} - 2\alpha \beta } \right\} + \left\{ {{{\left( {\alpha ' + \beta '} \right)}^2} - 2\alpha '\beta '} \right\}} \right] - 2\left( {\alpha + \beta } \right)\left( {\alpha ' + \beta '} \right)$
By the above relations we get
$ = 2\left[ {\left\{ {{{\left( p \right)}^2} - 2q} \right\} + \left\{ {{{\left( {p'} \right)}^2} - 2q'} \right\}} \right] - 2\left( {pp'} \right)$
$ = 2\left[ {{p^2} - 2q + p{'^2} - 2q' - pp'} \right]$
Hence the value of ${\left( {\alpha - \alpha '} \right)^2} + {\left( {\beta - \alpha '} \right)^2} + {\left( {\alpha - \beta '} \right)^2} + {\left( {\beta - \beta '} \right)^2}$ is$ = 2\left[ {{p^2} - 2q + p{'^2} - 2q' - pp'} \right]$.
Note: In this type of problem we tend to make mistakes in opening and closing the brackets of squaring and rooting. Always remember that for the quadratic polynomial \[a{x^2} + bx + c = 0\], the sum of the roots is \[ - \dfrac{b}{a}\] and the product of the roots is \[\dfrac{c}{a}\].
Given $\alpha ,\beta $ are the roots of the quadratic equation ${x^2} - px + q = 0$and $\alpha ',\beta '$are the roots of the quadratic equation${x^2} - p'x + q' = 0$.
So, for the quadratic equation ${x^2} - px + q = 0$
Sum of the roots is $\alpha + \beta = p$ and product of the roots is $\alpha \beta = q$
Similarly, for the quadratic equation ${x^2} - p'x + q' = 0$
Sum of the roots is \[\alpha ' + \beta ' = p'\] and product of the roots is $\alpha '\beta ' = q'$
Now consider ${\left( {\alpha - \alpha '} \right)^2} + {\left( {\beta - \alpha '} \right)^2} + {\left( {\alpha - \beta '} \right)^2} + {\left( {\beta - \beta '} \right)^2}$
\[ = {\alpha ^2} + \alpha {'^2} - 2\alpha \alpha ' + {\beta ^2} + \alpha {'^2} - 2\beta \alpha ' + {\alpha ^2} + \beta {'^2} - 2\alpha \beta ' + {\beta ^2} + \beta {'^2} - 2\beta \beta '\]
Grouping the terms, we have
$ = 2\left( {{\alpha ^2} + \alpha {'^2} + {\beta ^2} + \beta {'^2}} \right) - 2\left( {\alpha + \beta } \right)\left( {\alpha ' + \beta '} \right)$
$ = 2\left[ {\left\{ {{{\left( {\alpha + \beta } \right)}^2} - 2\alpha \beta } \right\} + \left\{ {{{\left( {\alpha ' + \beta '} \right)}^2} - 2\alpha '\beta '} \right\}} \right] - 2\left( {\alpha + \beta } \right)\left( {\alpha ' + \beta '} \right)$
By the above relations we get
$ = 2\left[ {\left\{ {{{\left( p \right)}^2} - 2q} \right\} + \left\{ {{{\left( {p'} \right)}^2} - 2q'} \right\}} \right] - 2\left( {pp'} \right)$
$ = 2\left[ {{p^2} - 2q + p{'^2} - 2q' - pp'} \right]$
Hence the value of ${\left( {\alpha - \alpha '} \right)^2} + {\left( {\beta - \alpha '} \right)^2} + {\left( {\alpha - \beta '} \right)^2} + {\left( {\beta - \beta '} \right)^2}$ is$ = 2\left[ {{p^2} - 2q + p{'^2} - 2q' - pp'} \right]$.
Note: In this type of problem we tend to make mistakes in opening and closing the brackets of squaring and rooting. Always remember that for the quadratic polynomial \[a{x^2} + bx + c = 0\], the sum of the roots is \[ - \dfrac{b}{a}\] and the product of the roots is \[\dfrac{c}{a}\].
Recently Updated Pages
Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

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

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE
