Question

If $\alpha, \beta$ are the roots of $x^2+7 x+9=0$, then the value of $\left[\left(\frac{\alpha}{3+\beta}\right)^2+\left(\frac{\beta}{3+\alpha}\right)^2\right]$ is
[Note: [k] denotes greatest integer function less than or equal to k.]

• A. 16
• B. 15
• C. 18
• D. 17

Answer 1

Answer: (D)

$ x^2+6 x+9=-x$
$(x+3)^2=-x$
$\Rightarrow \frac{\alpha^2}{(\beta+3)^2}+\frac{\beta^2}{(\alpha+3)^2}=-\left(\frac{\alpha^2}{\beta}+\frac{\beta^2}{\alpha}\right)$
$=-\left(\frac{(\alpha+\beta)^3-3 \alpha \beta(\alpha+\beta)}{\alpha \beta}\right)=\frac{154}{9} $
$\Rightarrow\left[\left(\frac{\alpha}{3+\beta}\right)^2+\left(\frac{\beta}{3+\alpha}\right)^2\right]=17$