Question

If $\alpha$and $\beta$are the roots of the equation $a{x}^2+bx+c=0$then find $\alpha /[ɑ\beta +b]+\beta /[aɑ+b]$

• A. $2/a$
• B. $2/b$
• C. $2/c$
• D. $-2/a$

Answer 1

The correct option is D

$-2/a$

Finding $\mathit{\alpha }\mathbf{}\mathbf{/}\mathbf{}\mathbf{[}\mathit{ɑ}\mathit{\beta }\mathbf{}\mathbf{+}\mathbf{}\mathit{b}\mathbf{]}\mathbf{}\mathbf{+}\mathbf{}\mathit{\beta }\mathbf{}\mathbf{/}\mathbf{}\mathbf{[}\mathit{a}\mathit{ɑ}\mathbf{}\mathbf{+}\mathbf{}\mathit{b}\mathbf{]}$:

Given If $\alpha$and $\beta$are the roots of the equation $a{x}^2+bx+c=0$

$$\begin{gathered} a{ɑ}^2+bɑ+c=0 \\ \Rightarrow aɑ+b=–c/ɑ.........\left(i\right) \\ ɑ{\beta }^2+b\beta +c=0 \\ \Rightarrow (ɑ\beta +b)=–c/\beta .........\left(ii\right) \end{gathered}$$

$$\begin{gathered} \alpha /[ɑ\beta +b]+\beta /[aɑ+b] \\ =(ɑ\beta /–c)+(ɑ\beta /–c) \\ =–2ɑ\beta /c \\ =-\frac{2{\displaystyle \frac{c}{a}}}{c} \\ =-\frac{2}{a} \end{gathered}$$

Hence, the correct option is D.