<Question>
If $\sqrt{3x-5}=2$, find all possible values of $x$.
</Question>

<think>
<Plan_1>
Okay, so I need to solve the equation √(3x - 5) = 2. Hmm, let me think about how to approach this. Alright, since the equation involves a square root, my first thought is to eliminate the square root by squaring both sides of the equation. That usually helps in these kinds of problems. Let me write that down.

<prompt_1.1> Square both sides of the equation √(3x - 5) = 2 to eliminate the square root. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
Starting with the original equation:
√(3x - 5) = 2

If I square both sides, that should get rid of the square root. Let me try that:
(√(3x - 5))² = 2²

Simplifying both sides gives:
3x - 5 = 4
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Now, I need to solve for x. Let me add 5 to both sides to isolate the term with x.

<prompt_2.1> Solve the equation 3x - 5 = 4 for x by isolating the term with x. </prompt_2.1>
-----

Wait, but sometimes when you square both sides of an equation, you can introduce extraneous solutions. So, I should check the solution in the original equation to make sure it works.

Let me verify.

<prompt_2.2> Verify the solution x = 3 by substituting it back into the original equation √(3x - 5) = 2. </prompt_2.2>

But just to be thorough, let me consider if there could be any other solutions. The original equation is a square root equal to 2. Since square roots are defined to be non-negative in real numbers, the right side is 2, which is positive, so there's no issue there.

<prompt_2.3> Consider if there could be any other solutions to the equation √(3x - 5) = 2 and confirm the solution. </prompt_2.3>
</Plan_2>

<Execution_2>
<execution_2.1>
3x - 5 + 5 = 4 + 5
Simplifies to:
3x = 9

Now, divide both sides by 3 to solve for x:
3x / 3 = 9 / 3
x = 3
</execution_2.1>

<execution_2.2>
If x = 3, then the left side of the original equation becomes √(3*3 - 5) = √(9 - 5) = √4 = 2. The right side is 2. So, 2 = 2.
</execution_2.2>

<execution_2.3>
So squaring both sides was a valid step because both sides are non-negative. Therefore, there shouldn't be any extraneous solutions here.

But just to double-check, let's consider if there's another way this could hold. Suppose there was another solution, but since the square root function is one-to-one (meaning if √a = √b, then a = b), there's only one solution here.
</execution_2.3>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Hold on, let me check that again.

<prompt_3.1> Re-evaluate the steps to solve the equation 3x - 5 = 4 to ensure the solution x = 3 is correct. </prompt_3.1>
-----

Based on execution_2.2, execution_2.3:
So, after all these steps and checks, I can confidently say that the solution is x = 3.
</Plan_3>

<Execution_3>
<execution_3.1>
If x is 3, then substituting back into the original equation:
√(3*3 - 5) = √(9 - 5) = √4 = 2, which matches the right side.
</execution_3.1>
</Execution_3>

<Final_answer>
Therefore, the solution to the equation $\sqrt{3x-5}=2$ is $x = \boxed{3}$.
</Final_answer>
</think>