The number of solutions of the equation
`sqrt(x^(2))-sqrt((x-1)^(2))+sqrt((x-2)^(2))=sqrt(5)` is

Explanation

To find the number of solutions for the equation

\[ \sqrt{x^2} - \sqrt{(x-1)^2} + \sqrt{(x-2)^2} = \sqrt{5} \]

we will analyze the equation step by step.

### Step 1: Rewrite the Equation
We can rewrite the equation using the properties of absolute values:

\[ |x| - |x-1| + |x-2| = \sqrt{5} \]

### Step 2: Consider Different Cases
Since the absolute value function behaves differently depending on the value of \(x\), we will consider different cases based on the critical points \(x = 0\), \(x = 1\), and \(x = 2\).

#### Case 1: \(x \geq 2\)
In this case, all absolute values can be removed as positive:

\[ x - (x - 1) + (x - 2) = \sqrt{5} \]
\[ x - x + 1 + x - 2 = \sqrt{5} \]
\[ x - 1 = \sqrt{5} \]
\[ x = \sqrt{5} + 1 \]

Since \(\sqrt{5} \approx 2.236\), we have \(x \approx 3.236\), which is valid for this case.

#### Case 2: \(1 \leq x < 2\)
In this case, we have:

\[ x - (x - 1) - (x - 2) = \sqrt{5} \]
\[ x - x + 1 - x + 2 = \sqrt{5} \]
\[ 3 - x = \sqrt{5} \]
\[ x = 3 - \sqrt{5} \]

Calculating \(3 - \sqrt{5} \approx 0.764\), which is valid for this case.

#### Case 3: \(0 \leq x < 1\)
Here, we have:

\[ x - (1 - x) - (2 - x) = \sqrt{5} \]
\[ x - 1 + x - 2 = \sqrt{5} \]
\[ 2x - 3 = \sqrt{5} \]
\[ 2x = \sqrt{5} + 3 \]
\[ x = \frac{\sqrt{5} + 3}{2} \]

Calculating \(\frac{\sqrt{5} + 3}{2} \approx 2.618\), which is not valid for this case since \(x\) must be less than 1.

#### Case 4: \(x < 0\)
In this case, all absolute values are negative:

\[ -x + (1 - x) + (2 - x) = \sqrt{5} \]
\[ -x + 1 - x + 2 - x = \sqrt{5} \]
\[ 3 - 3x = \sqrt{5} \]
\[ 3x = 3 - \sqrt{5} \]
\[ x = 1 - \frac{\sqrt{5}}{3} \]

Calculating \(1 - \frac{\sqrt{5}}{3} \approx 0.254\), which is not valid for this case since \(x\) must be less than 0.

### Step 3: Summary of Solutions
From the analysis of the four cases, we found valid solutions in:

- Case 1: \(x = \sqrt{5} + 1\) (valid)
- Case 2: \(x = 3 - \sqrt{5}\) (valid)
- Case 3: No valid solution
- Case 4: No valid solution

Thus, we have **two valid solutions** for the original equation.

### Conclusion
The number of solutions of the equation

\[ \sqrt{x^2} - \sqrt{(x-1)^2} + \sqrt{(x-2)^2} = \sqrt{5} \]

is **2**.

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