Answer
$x=-5$ and $x=2$
Work Step by Step
$x^{1/2}+3x^{-1/2}=10x^{-3/2}$
Rearrage the equation:
$x^{1/2}+\dfrac{3}{x^{1/2}}-10x^{-3/2}=0$
Multiply the whole equation by $x^{1/2}$:
$x^{1/2}\Big(x^{1/2}+\dfrac{3}{x^{1/2}}-10x^{-3/2}=0\Big)$
$x+3-10x^{-1}=0$
Rewrite $10x^{-1}$ as $\dfrac{10}{x}$ and multiply the whole equation by $x$:
$x+3-\dfrac{10}{x}=0$
$x\Big(x+3-\dfrac{10}{x}=0\Big)$
$x^{2}+3x-10=0$
Solve this equation by factoring:
$(x+5)(x-2)=0$
Set both factors equal to $0$ and solve each individual equation for $x$:
$x+5=0$
$x=-5$
$x-2=0$
$x=2$
The original equation is true for the two solutions found. The final answer is:
$x=-5$ and $x=2$
