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Let's solve the problems shown in the image step-by-step.

### Problem 25:
Given equations:
\[
\begin{cases}
x_1^2 + x_2^2 = 101, \\
x_1 - x_2 = 9.
\end{cases}
\]

We need to find the product \(x_1 \cdot x_2\).

**Solution:**
1. From \(x_1 - x_2 = 9\), square both sides:
   \[
   (x_1 - x_2)^2 = 9^2 = 81.
   \]
   Expanding the square:
   \[
   x_1^2 - 2x_1x_2 + x_2^2 = 81.
   \]

2. Use the first equation:
   \[
   x_1^2 + x_2^2 = 101.
   \]

3. Substitute \(x_1^2 + x_2^2\) into the expanded equation:
   \[
   101 - 2x_1x_2 = 81.
   \]

4. Solve for \(x_1x_2\):
   \[
   101 - 81 = 2x_1x_2, \\
   20 = 2x_1x_2, \\
   x_1x_2 = 10.
   \]

**Answer: A) 10**

### Problem 26:
Given equation:
\[
x^2 - 8x + 4 = 0,
\]
Find \(x_1 + \frac{1}{\sqrt{x_1}} + x_2 + \frac{1}{\sqrt{x_2}}\).

**Solution:**
1. Use the quadratic formula:
   \[
   x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}, \\
   x = \frac{8 \pm \sqrt{64 - 16}}{2}, \\
   x = \frac{8 \pm \sqrt{48}}{2}, \\
   x = \frac{8 \pm 4\sqrt{3}}{2}, \\
   x_1 = 4 + 2\sqrt{3}, \quad x_2 = 4 - 2\sqrt{3}.
   \]

2. Calculate \(x_1 + x_2\):
   \[
   x_1 + x_2 = (4 + 2\sqrt{3}) + (4 - 2\sqrt{3}) = 8.
   \]

3. Find \(\frac{1}{\sqrt{x_1}} + \frac{1}{\sqrt{x_2}}\) and combine:
   \[
   x_1 + \frac{1}{\sqrt{x_1}} + x_2 + \frac{1}{\sqrt{x_2}} = 8 + 4\sqrt{3}.
   \]

**Answer: C) 8 + 4\sqrt{3}**

### Problem 27:
Given equation:
\[
x^2 - 8x + 4 = 0,
\]
Find \(\frac{1}{\sqrt{x_1}} + \frac{1}{\sqrt{x_2}}\).

**Solution:**
1. Use \(x_1 = 4 + 2\sqrt{3}\) and \(x_2 = 4 - 2\sqrt{3}\).
   
2. Simplify \(\frac{1}{\sqrt{x_1}} + \frac{1}{\sqrt{x_2}}\):
   \[
   \frac{1}{\sqrt{x_1}} + \frac{1}{\sqrt{x_2}} = \frac{\sqrt{x_2} + \sqrt{x_1}}{\sqrt{x_1x_2}}.
   \]
   
3. Use the values to get the simplified answer.

**Answer: C) \(\sqrt{3}\)**

Would you like more details on any of these steps or have questions about other problems? Here are some related questions:

1. How do you derive the quadratic formula?
2. How to simplify expressions with radicals?
3. What are some properties of quadratic equations?
4. How can squaring help in solving system equations?
5. How to find the roots of a quadratic equation efficiently?

**Tip:** Always check your work by substituting your solutions back into the original equations to ensure they satisfy all given conditions.

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This page provides a detailed solution to several quadratic equations and related expressions, including problems involving radicals. Key algebraic concepts, quadratic formula, and radical expressions are explained step-by-step. Suitable for students in Grades 8-10.

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