College Algebra, Chapter 1, 1.5, Section 1.5, Problem 30

Find all real solutions of the equation $\displaystyle x - \sqrt{9 - 3x} = 0$


$
\begin{equation}
\begin{aligned}

x - \sqrt{9 - 3x} =& 0
&& \text{Given}
\\
\\
x =& \sqrt{9 - 3x}
&& \text{Add } \sqrt{9 - 3x}
\\
\\
(x)^2 =& (\sqrt{9 - 3x})^2
&& \text{Square both sides}
\\
\\
x^2 =& 9 - 3x
&& \text{Add } 3x
\\
\\
x^2 + 3x =& 9
&& \text{Complete the square: add } \left( \frac{3}{2} \right)^2 = \frac{9}{4}
\\
\\
x^2 + 3x + \frac{9}{4} =& 9 + \frac{9}{4}
&& \text{Perfect square}
\\
\\
\left(x + \frac{3}{2} \right)^2 =& \frac{45}{4}
&& \text{Take the square root}
\\
\\
x + \frac{3}{2} =& \pm \sqrt{\frac{45}{4}}
&& \text{Subtract } \frac{3}{2} \text{ and simplify}
\\
\\
x =& \frac{-3}{2} \pm \frac{3 \sqrt{5 2}}{}
&& \text{Evaluate}
\\
\\
x =& \frac{-3 + 3 \sqrt{5}}{2} \text{ and } x = \frac{-3 - 3 \sqrt{5}}{2}
&& \text{Solve for } x


\end{aligned}
\end{equation}
$