College Algebra, Chapter 1, 1.5, Section 1.5, Problem 50

Find all real solutions of the equation $\displaystyle x^{\frac{1}{2}} + 3x^{\frac{-1}{2}} = 10x^{\frac{-3}{2}}$


$
\begin{equation}
\begin{aligned}

x^{\frac{1}{2}} + 3x^{\frac{-1}{2}} =& 10x^{\frac{-3}{2}}
&& \text{Given}
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x^{\frac{1}{2}} + 3x^{\frac{-1}{2}} - 10x^{\frac{-3}{2}} =& 0
&& \text{Subtract } 10x^{\frac{-3}{2}}
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x^{\frac{1}{2}} + \frac{3}{x^{\frac{1}{2}}} - \frac{10}{x^{\frac{3}{2}}} =& 0
&& \text{Multiply both sides by } x^{\frac{3}{2}}
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x^{\frac{1}{2}} \cdot x^{\frac{3}{2}} + \frac{3x^{\frac{3}{2}}}{x^{\frac{1}{2}}} - 10 =& 0
&& \text{Use the Properties of Exponents}
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x^2 + 3x - 10 =& 0
&& \text{Factor}
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(x + 5)(x - 2) =& 0
&& \text{Zero Product Property}
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x + 5 =& 0 \text{ and } x - 2 = 0
&& \text{Solve for } x
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x =& -5 \text{ and } x = 2
&&
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x =& 2
&& \text{The only solution that satisfy the equation}

\end{aligned}
\end{equation}
$