Question

# 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${\left( {\alpha - \alpha '} \right)^2} + {\left( {\beta - \alpha '} \right)^2} + {\left( {\alpha - \beta '} \right)^2} + {\left( {\beta - \beta '} \right)^2}$

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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 $\dfrac{c}{a}$.