The problem in the image is asking to demonstrate that for all $(x, y) \in \mathbb{R}^2$:

$2 \sqrt{x - 1 + 4 \sqrt{x - 4}} = x + y \iff (x = 2 \text{ and } x = 8)$

Let's break this down and solve it step-by-step:

Given equation:

$2 \sqrt{x - 1 + 4 \sqrt{x - 4}} = x + y$

We need to show that this equation holds if and only if $x = 2$ and $x = 8$.

Step 1: Simplifying the left-hand side

The equation starts with a nested square root: $2 \sqrt{x - 1 + 4 \sqrt{x - 4}} = x + y$ First, consider simplifying the innermost square root, i.e., $\sqrt{x - 4}$.

We want to find the values of $x$ for which this expression is defined and satisfies the equation.

Step 2: Analyze the condition

For the square roots to be valid, $x - 4 \geq 0$, which implies: $x \geq 4$ Now, we'll test specific values of $x$ mentioned in the problem, $x = 2$ and $x = 8$.

Case 1: $x = 2$

For $x = 2$, the innermost square root, $\sqrt{x - 4}$, would give $\sqrt{2 - 4} = \sqrt{-2}$, which is not real. Therefore, $x = 2$ is not a valid solution.

Case 2: $x = 8$

For $x = 8$, we substitute into the equation:

$2 \sqrt{8 - 1 + 4 \sqrt{8 - 4}} = 8 + y$ Simplify the square roots: $2 \sqrt{7 + 4 \sqrt{4}} = 8 + y$ $\sqrt{4} = 2, \quad \text{so the equation becomes} \quad 2 \sqrt{7 + 8} = 8 + y$ $2 \sqrt{15} = 8 + y$ Therefore, for $x = 8$, $y = 2 \sqrt{15} - 8$.

Conclusion:

The equation holds for $x = 8$ and not for $x = 2$ based on the analysis. However, it seems the original question implies showing that the given equation leads to these values of $x = 2$ and $x = 8$ as special conditions.

Would you like to explore further details or have questions about this solution?

Here are 5 follow-up questions to expand on the concept:

1. Why is $x = 2$ not a valid solution for the given equation?
2. How can we verify whether other values of $x$ could also satisfy the equation?
3. What is the significance of the nested square roots in the problem?
4. Could there be other methods to solve this problem algebraically?
5. What does the condition $x = 2 \text{ and } x = 8$ imply about the nature of the equation?

Tip: When solving equations with nested square roots, always check for the domain restrictions, as square roots require non-negative values.