If $α$ and $β$ are the zeros of the quadratic polynomial $f(x)\ =\ ax^2\ +\ bx\ +\ c$, then evaluate:
$\frac{β}{aα\ +\ b}\ +\ \frac{α}{aβ\ +\ b}$

Given:

$α$  and  $β$  are the zeros of the quadratic polynomial  $f(x)\ =\ ax^2\ +\ bx\ +\ c$.

To do:

We have to find the value of $\frac{β}{aα\ +\ b}\ +\ \frac{α}{aβ\ +\ b}$.

Solution:

The given quadratic equation is $ax^2+bx+c=0$, where a, b and c are constants and $a≠0$.

Sum of the roots $= α+β = \frac{-b}{a}$.

Product of the roots $= αβ = \frac{c}{a}$.

We know that,

$ \begin{array}{l}
\frac{\beta }{a\alpha +b} +\frac{\alpha }{a\beta +b} =\frac{\beta ( a\beta +b) +\alpha ( a\alpha +b)}{( a\alpha +b)( a\beta +b)}\\
\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{a\beta ^{2} +b\beta +a\alpha ^{2} +b\alpha }{a^{2} \alpha \beta +ab\alpha +ab\beta +b^{2}}\\
\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{a\left( \alpha ^{2} +\beta ^{2}\right) +b( \alpha +\beta )}{a^{2} \alpha \beta +ab( \alpha +\beta ) +b^{2}}\\
\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{a\left(( \alpha +\beta )^{2} -2\alpha \beta \right) +b( \alpha +\beta )}{a^{2} \alpha \beta +ab( \alpha +\beta ) +b^{2}}\\
\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{a\left(\left(\frac{-b}{a}\right)^{2} -2\left(\frac{c}{a}\right)\right) +b\left(\frac{-b}{a}\right)}{a^{2}\left(\frac{c}{a}\right) +ab\left( -\frac{b}{a}\right) +b^{2}} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\
\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{a\left(\frac{b^{2}}{a^{2}} -\frac{2c}{a}\right) -\frac{b^{2}}{a}}{ac-b^{2} +b^{2}}\\
\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{a\left(\frac{b^{2} -2ac}{a^{2}}\right) -\frac{b^{2}}{a}}{ac}\\
\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{\left(\frac{b^{2} -2ac-b^{2}}{a}\right)}{ac}\\
\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{-2ac}{a^{2} c}\\
\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{-2}{a}
\end{array}$

The value of $\frac{β}{aα\ +\ b}\ +\ \frac{α}{aβ\ +\ b}$ is $-\frac{2}{a}$. 

Tutorialspoint

Updated on: 10-Oct-2022

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