4. Solve: $$ \quad x=\sqrt{x+6} $$

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Solve the equation $$ x=\sqrt{x+6} $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$x=\sqrt{x+6}$$
- step1: Find the domain:
  $$x=\sqrt{x+6},x\geq -6$$
- step2: Swap the sides:
  $$\sqrt{x+6}=x$$
- step3: Evaluate:
  $$\sqrt{x+6}=x,x\geq 0$$
- step4: Solve the equation:
  $$\begin{align}&x=3\&x=-2\end{align},x\geq 0$$
- step5: Find the intersection:
  $$x=3$$
- step6: Check if the solution is in the defined range:
  $$x=3,x\geq -6$$
- step7: Find the intersection:
  $$x=3$$
- step8: Check the solution:
  $$x=3$$
The solution to the equation $$x=\sqrt{x+6}$$ is $$x=3$$.
$$ x=3 $$

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