The extracted text appears to be empty. I will attempt to enhance the image and extract the text again.It appears that text extraction from the image is still not yielding results. Let's proceed by manually analyzing the image. Please hold on for a moment while I review the image content visually.

---

The image contains mathematical questions involving quadratic numbers and expressions. Here are the provided questions with their details:

1. Question 1: $\text{Determine whether the number } \sqrt{2} + \sqrt{3} \text{ is rational or irrational.}$

   • Answer: The number $\sqrt{2} + \sqrt{3}$ is irrational.

2. Question 2: $\text{Consider the equation } x^2 + y^2 = 7. \text{ Is there a pair of rational numbers } (x, y) \text{ that satisfy this equation?}$

   • Answer: There are no rational numbers $(x, y)$ that satisfy $x^2 + y^2 = 7$.

3. Question 3: $\text{Prove that } \sqrt{2} + \sqrt{3} \text{ is irrational.}$

   • Explanation: $\text{Assume } \sqrt{2} + \sqrt{3} \text{ is rational. Let } \sqrt{2} + \sqrt{3} = r \text{ where } r \text{ is rational.}$ $\text{Then } \sqrt{2} = r - \sqrt{3}. \text{ Squaring both sides:}$ $2 = r^2 - 2r\sqrt{3} + 3.$ $\Rightarrow 2r\sqrt{3} = r^2 + 1.$ $\Rightarrow \sqrt{3} = \frac{r^2 + 1}{2r}.$ $\text{This implies } \sqrt{3} \text{ is rational, which is a contradiction. Therefore, } \sqrt{2} + \sqrt{3} \text{ is irrational.}$

4. Question 4: $\text{Consider the quadratic equation } x^2 - (3+\sqrt{2})x + (2+\sqrt{2}) = 0. \text{ Determine if the roots are rational or irrational.}$

   • Answer: The roots of the equation are irrational.

---

Would you like further details or explanations on any of these questions?

Additional Questions:

1. Why is $\sqrt{2} + \sqrt{3}$ considered irrational?
2. How can we systematically determine if a given number is rational or irrational?
3. What methods are used to determine the rationality of the roots of a quadratic equation?
4. Can the rational root theorem be applied to non-integer coefficients?
5. How do properties of square-free integers affect the rationality of expressions?
6. Why is the equation $x^2 + y^2 = 7$ unsolvable in rational numbers?
7. What are the implications of having a square-free integer in quadratic expressions?
8. Can complex numbers be considered in the context of these questions?

Tip:

When determining whether a number is rational or irrational, always consider using contradiction by assuming the number is rational and showing that this assumption leads to an inconsistency.