Question

If $\alpha ,\beta$ are the roots of the equations $x^2+4x+3=0$, then the equation whose roots are $2\alpha +\beta$ and $\alpha +2\beta$ is

• A. $x^2-12x-33=0$
• B. $x^2-12x+35=0$
• C. $x^2+12x-33=0$
• D. $x^2+12x+35=0$

Answer 1

The correct option is D

$x^2+12x+35=0$

Explanation for the correct option:

Find the equation:

Given that,

$\alpha ,\beta$ are the roots of the equations $x^2+4x+3=0$, then

Sum of root$=-\frac{\mathrm{coefficient}\mathrm{of}x}{\mathrm{coefficient}\mathrm{of}{x}^2}$

$\begin{array}{rcl}& \Rightarrow & \alpha +\beta =\frac{-4}{1}\\ & =& -4...\left(1\right)\end{array}$

Product of root$=\frac{\mathrm{constant}\mathrm{term}}{\mathrm{coefficient}\mathrm{of}{x}^2}$

$\begin{array}{rcl}& \Rightarrow & \alpha ·\beta =\frac{3}{1}\\ & =& 3...\left(2\right)\end{array}$

Here, $2\alpha +\beta$ and $\alpha +2\beta$ are the roots of the required equation, so

The required equation is $x^2-\left(\mathrm{Sum}\mathrm{of}\mathrm{root}\right)x+\left(\mathrm{Product}\mathrm{of}\mathrm{root}\right)=0$

Now,

The sum of root $=2\alpha +\beta +\alpha +2\beta$

$\begin{array}{rcl}& \Rightarrow & 2\alpha +\beta +\alpha +2\beta =3\left(\alpha +\beta \right)\\ & =& 3\times \left(-4\right)\left[\mathrm{from}\left(1\right)\right]\\ & =& -12\end{array}$

And

The product of root$=\left(2\alpha +\beta \right)\left(\alpha +2\beta \right)$

$\begin{array}{rcl}& \Rightarrow & \left(2\alpha +\beta \right)\left(\alpha +2\beta \right)=2{\alpha }^2+4\beta \alpha +\beta \alpha +2{\beta }^2\\ & =& 2{\alpha }^2+2{\beta }^2+4\alpha \beta +\beta \alpha \\ & =& 2\left({\alpha }^2+{\beta }^2+2\alpha \beta \right)+\beta \alpha \left[\because a^2+b^2+2ab={\left(a+b\right)}^2\right]\\ & =& 2{\left(\alpha +\beta \right)}^2+\alpha \beta \\ & =& 2{\left(-4\right)}^2+3\mathbf{}\left[\mathrm{from}\left(1\right)\&\left(2\right)\right]\\ & =& 2\times 16+3\\ & =& 35\end{array}$

By substituting these values in the required equation, we get

$$\begin{gathered} \Rightarrow x^2-\left(-12\right)x+35=0 \\ \Rightarrow x^2+12x+35=0 \end{gathered}$$

Hence, the correct answer is D.