What is the sum of the squares of the roots of the equation `x^(2)-7[x] + 5 = 0` ? (Here [x] denotes the greatest integer less than or equal to x. For example [3.4] = 3 and [-2.3] =-3).

Answer

To find the sum of the squares of the roots of the equation \(x^2 - 7[x] + 5 = 0\), where \([x]\) denotes the greatest integer less than or equal to \(x\), we will follow these steps:

### Step 1: Rewrite the equation
The given equation is:
\[ x^2 - 7[x] + 5 = 0 \]
We can rearrange this to express \(x^2\):
\[ x^2 = 7[x] - 5 \]

### Step 2: Analyze the inequality
Since \(x^2\) is always non-negative, we have:
\[ 7[x] - 5 \geq 0 \implies 7[x] \geq 5 \implies [x] \geq \frac{5}{7} \]
Since \([x]\) is an integer, we conclude:
\[ [x] \geq 1 \]

### Step 3: Determine possible values for \([x]\)
Since \([x]\) must be a non-negative integer, we can consider values starting from 1. We also know that \(x\) must be less than \(7\) because \(x^2 < 7x\) implies \(x(x-7) < 0\), which gives \(0 < x < 7\). Thus, \([x]\) can take values from 1 to 6.

### Step 4: Solve for each integer value of \([x]\)
1. **For \([x] = 1\)**:
\[ x^2 = 7(1) - 5 = 2 \implies x = \sqrt{2} \]
\(\sqrt{2} \in [1, 2)\) so it is valid.

2. **For \([x] = 2\)**:
\[ x^2 = 7(2) - 5 = 9 \implies x = 3 \]
\(3 \notin [2, 3)\) so it is invalid.

3. **For \([x] = 3\)**:
\[ x^2 = 7(3) - 5 = 16 \implies x = 4 \]
\(4 \notin [3, 4)\) so it is invalid.

4. **For \([x] = 4\)**:
\[ x^2 = 7(4) - 5 = 23 \implies x = \sqrt{23} \]
\(\sqrt{23} \in [4, 5)\) so it is valid.

5. **For \([x] = 5\)**:
\[ x^2 = 7(5) - 5 = 30 \implies x = \sqrt{30} \]
\(\sqrt{30} \in [5, 6)\) so it is valid.

6. **For \([x] = 6\)**:
\[ x^2 = 7(6) - 5 = 37 \implies x = \sqrt{37} \]
\(\sqrt{37} \in [6, 7)\) so it is valid.

### Step 5: List the valid roots
The valid roots we found are:
- \( \sqrt{2} \)
- \( \sqrt{23} \)
- \( \sqrt{30} \)
- \( \sqrt{37} \)

### Step 6: Calculate the sum of the squares of the roots
Now we calculate the sum of the squares of these roots:
\[ (\sqrt{2})^2 + (\sqrt{23})^2 + (\sqrt{30})^2 + (\sqrt{37})^2 = 2 + 23 + 30 + 37 \]
Calculating this gives:
\[ 2 + 23 = 25 \]
\[ 25 + 30 = 55 \]
\[ 55 + 37 = 92 \]

### Final Answer
Thus, the sum of the squares of the roots is:
\[ \boxed{92} \]