grenivkah3z

2022-07-04

If $x-\sqrt{\frac{2}{x}}=3$ then $x-\sqrt{2x}=?$

Sariah Glover

Expert

$\begin{array}{rl}& x-\sqrt{\frac{2}{x}}=3\\ & x\sqrt{x}-\sqrt{2}=3\sqrt{x}\\ & \sqrt{x}\left(x-3\right)=\sqrt{2}\\ & x\left(x-3{\right)}^{2}=2\\ & \left(x-2\right)\left({x}^{2}-4x+1\right)=0\end{array}$
Note that $x=2$ is not a solution, so

The second is not a solution either, so $x=2+\sqrt{3}.$
The second is not a solution either, so $x=2+\sqrt{3}.$
In that case, $x-\sqrt{2x}=2+\sqrt{3}-\sqrt{2\left(2+\sqrt{3}\right)}=2+\sqrt{3}-\sqrt{\left(\sqrt{3}+1{\right)}^{2}}=1.$

Kristen Stokes

Expert

Let $\sqrt{\frac{x}{2}}=y\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}x=2{y}^{2}$
$\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}2{y}^{2}-\frac{1}{y}=3\phantom{\rule{thickmathspace}{0ex}}⟺\phantom{\rule{thickmathspace}{0ex}}2{y}^{3}-3y-1=0$
Let

(1),(2) will represent the same equation iff
$\frac{1}{2}=\frac{-1}{2a-3}=\frac{a}{1}$

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