[tex] \Large \mathcal{SOLUTION:} [/tex]
[tex] \begin{array}{l} \:\: \quad\qquad \sqrt{x + 6} +3 = \sqrt{x + 9} \\ \\ \textsf{Square both sides.} \\ \\ \quad \begin{array}{c} \left(\sqrt{x + 6} +3\right)^2 = \left(\sqrt{x + 9}\right)^2 \\ \\ \cancel{x} + 6 + 2(3)\sqrt{x + 6} + \cancel{9} = \cancel{x} + \cancel{9} \end{array} \\ \\ \textsf{Cancel out like terms then subtract 6 from both} \\ \textsf{sides.} \\ \\ \qquad \qquad \quad 6\sqrt{x + 6} = -6 \\ \\ \textsf{Divide both sides of the equation by 6.} \\ \\ \quad\quad\quad \qquad \sqrt{x + 6} = -1 \\ \\ \textsf{Square both sides again.} \\ \\ \qquad\quad\:\begin{array}{c} \left(\sqrt{x + 6}\right)^2 = (-1)^2 \\ \\ x + 6 = 1 \\ \\ x = 1 - 6 \\ \\ x = -5 \end{array} \\ \\ \textsf{Verifying,} \\ \\ \begin{aligned} \:\: \quad\qquad \sqrt{- 5 + 6} +3 &= \sqrt{-5 + 9} \\ \\ 1 +3 &= \sqrt{4} \\ \\ 4 &= 2 \quad \textsf{(False)} \end{aligned} \\ \\ \therefore \boxed{\bold{No\ Solution}} \end{array} [/tex]
