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

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


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