Question

If the roots of the equation $3x^2+9x+2=0$ are in the ratio $m:n$, then find $\sqrt{\frac{m}{n}}+\sqrt{\frac{n}{m}}$

• A. $\frac{3\sqrt{3}}{\sqrt{2}}$
• B. $3\sqrt{2}$
• C. $\frac{\sqrt{2}}{\sqrt{3}}$
• D. $2\sqrt{3}$

Answer 1

The correct option is A $\frac{3\sqrt{3}}{\sqrt{2}}$

${3x}^2+9x+2=0$

let roots be $m\alpha \&n\alpha$

sum = $m\alpha +n\alpha =(m+n)\alpha =\frac{-9}{3}=-3$

product = $m\alpha \times n\alpha =mn{\alpha }^2=\frac{2}{3}=\sqrt{\frac{m}{n}}+\sqrt{\frac{n}{m}}=\frac{m+n}{\sqrt{nm}}=\frac{\frac{-3}{\alpha }}{\sqrt{\frac{2}{3{\alpha }^2}}}=\frac{3}{\alpha }\times \alpha \sqrt{\frac{3}{2}}=\frac{3\sqrt{3}}{\sqrt{2}}$

as m, n are positive

${\alpha }^2$ comes out of square root as $-\alpha$