College Algebra, Chapter 1, 1.5, Section 1.5, Problem 34

Find all real solutions of the equation $\displaystyle x + 2 \sqrt{x - 7} = 10$


$
\begin{equation}
\begin{aligned}

x + 2 \sqrt{x - 7} =& 10
&& \text{Given}
\\
\\
2 \sqrt{x - 7} =& 10 - x
&& \text{Subtract } x
\\
\\
(2 \sqrt{x - 7})^2 =& (10 - x)^2
&& \text{Square both sides}
\\
\\
4(x - 7) =& 100 - 20x + x^2
&& \text{Use FOIL method}
\\
\\
4x - 28 =& 100 - 20x + x^2
&& \text{Combine like terms}
\\
\\
x^2 - 24x + 128 =& 0
&& \text{Factor out}
\\
\\
(x - 8)(x - 16) =& 0
&& \text{Zero Product Property}
\\
\\
x - 8 =& 0 \text{ and } x - 16 = 0
&& \text{Solve for } x
\\
\\
x =& 8 \text{ and } x = 16
&&
\\
\\
x =& 8
&& \text{The only solution that satisfy the equation } x + 2 \sqrt{x - 7} = 10

\end{aligned}
\end{equation}
$

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