Question

If $\alpha$ and $\beta$ are the roots of the equation $2x^2-3x-6=0$, then the equation whose roots are ${\alpha }^2+2,{\beta }^2+2$, is

• A. $4x^2+49x+118=0.$
• B. $4x^2-49x+118=0.$
• C. $4x^2-49x-118=0.$
• D. None of these

Answer 1

The correct option is B $4x^2-49x+118=0.$

Given, $\alpha$ and $\beta$ are the roots of the equation $2x^2-3x-6=0$
$\alpha +\beta =3/2,\alpha \beta =-6/2=-3$
For equation whose roots are ${\alpha }^2+2,{\alpha }^2+2$
Sum of roots $=S={\alpha }^2+{\beta }^2+4$
$={(\alpha +\beta )}^2-2\alpha \beta +4$
$=49/4$
Products of root $P={\alpha }^2{\beta }^2+2({\alpha }^2+{\beta }^2)+4$
$={\alpha }^2{\beta }^2+4+2[{(\alpha +\beta )}^2-2\alpha \beta ]$
$=\frac{118}{4}$
The equation is $x^2-Sx+P=0$
$=>x^2\left(49/4\right)x+\left(118/4\right)=0$
$=>4x^2-49x+118=0.$