We will solve the equation:

\[
\sqrt{x+2} + \sqrt{x-1} = 1
\]

### Step 1: Isolate one of the square roots
We can isolate one of the square roots, for example, \(\sqrt{x+2}\):

\[
\sqrt{x+2} = 1 - \sqrt{x-1}
\]

### Step 2: Square both sides to eliminate the square root
Now, square both sides of the equation to remove the square root on the left-hand side:

\[
(\sqrt{x+2})^2 = (1 - \sqrt{x-1})^2
\]

Simplifying both sides:

\[
x + 2 = 1 - 2\sqrt{x-1} + (x-1)
\]

### Step 3: Simplify the equation
Now simplify the right-hand side:

\[
x + 2 = x + 1 - 2\sqrt{x-1}
\]

### Step 4: Isolate the remaining square root
Subtract \(x + 1\) from both sides:

\[
2 = -2\sqrt{x-1}
\]

Divide by \(-2\):

\[
-1 = \sqrt{x-1}
\]

### Step 5: Check for contradictions
Since a square root cannot be negative, this leads to a contradiction. Therefore, there are **no real solutions** to this equation.

Would you like more details or have any questions?

Here are 5 related questions for further practice:

1. Solve \(\sqrt{x+3} + \sqrt{x-2} = 2\).
2. Solve \(\sqrt{x-1} + \sqrt{2x+1} = 3\).
3. Solve \( \sqrt{2x} + \sqrt{x-3} = 4 \).
4. Solve \(\sqrt{3x+1} - \sqrt{x+4} = 1\).
5. Solve \(\sqrt{x+6} = \sqrt{x} + 2\).

**Tip:** When solving square root equations, always check for extraneous solutions by substituting the final answers back into the original equation!

This step-by-step guide explains how to solve the square root equation sqrt(x+2) + sqrt(x-1) = 1. Explore algebraic techniques, such as isolating square roots, squaring both sides, and identifying contradictions in square root equations. Ideal for high school students in Grades 9-12.

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