Question

Solve the following equations:
$5\sqrt{\frac{3}{x}}+7+\sqrt{\frac{x}{3}}=22\frac{2}{3}$.

Answer 1

$5\sqrt{\frac{3}{x}}+7\sqrt{\frac{x}{3}}=22\frac{2}{3}$

Put $t=\sqrt{\frac{3}{x}}$ then $\sqrt{\frac{x}{3}}=\frac{1}{t}$

$5t+\frac{7}{t}=\frac{68}{3}\frac{5t^2+7}{t}=\frac{68}{3}15t^2+21=68t15t^2-68t+21=015t^2-5t-63t+21=05t(3t-1)-21(3t-1)=0(5t-21)(3t-1)=0t=\frac{21}{5},\frac{1}{3}$

Now , $t=\sqrt{\frac{3}{x}}$

$t^2=\frac{3}{x}x=\frac{3}{t^2}$

For $t=\frac{21}{5}$

$x=\frac{3}{{\left(\frac{21}{5}\right)}^2}=\frac{25}{147}$

For $t=\frac{1}{3}$

$x=\frac{3}{{\left(\frac{1}{3}\right)}^2}=27$

$\therefore x=27,\frac{25}{147}$