Question

If $\alpha$and $\beta$are the roots of$x^2+5x+4=0$, then the equation, whose roots are$\frac{\left(\alpha +2\right)}{3}$ and$\frac{\left(\beta +2\right)}{3}$, is

• A. $9x^2+3x+2=0$
• B. $9x^2-3x-2=0$
• C. $9x^2+3x-2=0$
• D. $9x^2-3x+2=0$

Answer 1

The correct option is D

$9x^2-3x+2=0$

Explanation for correct option:

Find the equation:

Since $\alpha$and $\beta$are the roots of$x^2+5x+4=0$

So, $\alpha +\beta =-5$and $\alpha \beta =4$

Now the roots of the given equation are$\frac{\left(\alpha +2\right)}{3}$ and$\frac{\left(\beta +2\right)}{3}$

$$\begin{gathered} \frac{\left(\alpha +2\right)}{3}+\frac{\left(\beta +2\right)}{3}=\frac{\left(\alpha +\beta +4\right)}{3} \\ =\frac{-5+4}{3} \\ =\frac{-1}{3} \end{gathered}$$

$$\begin{gathered} \frac{\left(\alpha +2\right)}{3}\times \frac{\left(\beta +2\right)}{3}=\frac{\left[\alpha \beta +2\left(\alpha +\beta \right)+4\right]}{9} \\ =\frac{\left[4-10+4\right]}{9} \\ =\frac{-2}{9} \end{gathered}$$

Thus the required equation is

$$\begin{gathered} x^2+(1/3)x–(2/9)=0 \\ \Rightarrow 9x^2–3x+2=0 \end{gathered}$$

Hence, the correct option is D.