solve:

x = 4 is the answer, if you take the positive square roots. There may be other answers if you take the negative square root on one or both sides.

I got that by trial and error, not by using algebra. The algebra got too messy.

Whatever roots exist must be between x = 0 and 5, otherwise you get imaginary numbers. As I said in a previous post, x=4 is the answer is you take postive squate roots on both sides. x=1 is also an answer, if you take the positive root of 1 on the left and the negative root of 4 on the right.

Here is the algebraic proof:

sqrt(x) + 2 = sqrt(5-x) + 3
sqrt(x) -1 = sqrt(5-x)
2x = 2 sqrt(x) +4
x = sqrt(x) + 2
Let sqrt(x) = y, and you have a quadratic equation:
y^2 -y +2 = (y-2)(y+1) = 0
y = 2 or -1
x = y^2 = 4 or 1

To solve the given equation, we need to isolate the variable x. Let's break down the process step by step.

1. Start by subtracting 2 from both sides of the equation to eliminate the +2 term on the left side:
sqrt(x) = sqrt(5 - x) + 1

2. Next, square both sides of the equation to eliminate the square root on the left side:
(sqrt(x))^2 = (sqrt(5 - x) + 1)^2

Squaring the left side gives us:
x = (sqrt(5 - x) + 1)^2

To simplify the equation further, we need to square the expression on the right side:

3. Expand the expression (sqrt(5 - x) + 1)^2 using the formula (a + b)^2 = a^2 + 2ab + b^2:
x = (sqrt(5 - x))^2 + 2(sqrt(5 - x))(1) + 1^2

4. Simplify the equation:
x = 5 - x + 2(sqrt(5 - x)) + 1

5. Rearrange the terms:
2x = 6 + 2(sqrt(5 - x))

6. Subtract 2(sqrt(5 - x)) from both sides:
2x - 2(sqrt(5 - x)) = 6

7. Factor out 2:
2(x - sqrt(5 - x)) = 6

8. Divide both sides by 2 to isolate the expression (x - sqrt(5 - x)):
(x - sqrt(5 - x)) = 3

9. Lastly, add sqrt(5 - x) to both sides of the equation:
x = sqrt(5 - x) + 3

Now, we have another equation. To solve for x, we need to follow a similar process:

1. Subtract 3 from both sides to eliminate the +3 term on the right side:
x - 3 = sqrt(5 - x)

2. Square both sides to eliminate the square root on the right side:
(x - 3)^2 = (sqrt(5 - x))^2

3. Expand and simplify:
x^2 - 6x + 9 = 5 - x

4. Rearrange the terms:
x^2 - 5x + 4 = 0

5. Factor the equation:
(x - 4)(x - 1) = 0

By setting each factor equal to 0, we can solve for x:

x - 4 = 0 --> x = 4
x - 1 = 0 --> x = 1

Therefore, the solutions to the equation are x = 4 and x = 1.