In this aptitude (simplification and quadratic equations) question, solve the equation: (x - 2)^2 - 36 = 0, find the possible real values of x, and then choose the value of x that belongs to the set of natural numbers (x in N).

Introduction / Context:
This question involves a simple quadratic equation written in shifted square form. The candidate must solve for x, identify all real solutions, and then select the solution that is a natural number. It reinforces basic algebraic manipulation and understanding of number sets such as integers and natural numbers.

Given Data / Assumptions:
We are given the equation (x - 2)^2 - 36 = 0.
We know x belongs to the natural numbers, so x must be a positive integer.
We must find all real solutions and then filter those that are natural numbers.

Concept / Approach:
The equation is of the form (x - a)^2 - b^2 = 0, which resembles a difference of squares. We can either expand and solve as a standard quadratic or directly interpret it as (x - 2)^2 = 36. Taking square roots gives two real solutions, because both positive and negative roots are possible. Finally, we use the natural number condition to pick the appropriate value.

Step-by-Step Solution:
Start with (x - 2)^2 - 36 = 0. Rearrange to (x - 2)^2 = 36. Take square roots on both sides: x - 2 = 6 or x - 2 = -6. From x - 2 = 6 we get x = 8. From x - 2 = -6 we get x = -4. So the real solutions are x = 8 and x = -4.

Verification / Alternative check:
Substitute x = 8 into the original equation: (8 - 2)^2 - 36 = 6^2 - 36 = 36 - 36 = 0, so x = 8 is valid. Substitute x = -4: (-4 - 2)^2 - 36 = (-6)^2 - 36 = 36 - 36 = 0, so x = -4 is also valid as a real root. However we must respect the condition that x is a natural number, which usually means positive integers 1, 2, 3, and so on, so we keep only x = 8.

Why Other Options Are Wrong:
Options -4 and -8 are not natural numbers because they are negative. Option 4 fails because (4 - 2)^2 - 36 = 4 - 36 = -32, which is not zero. Option 2 also fails because (2 - 2)^2 - 36 = 0 - 36 = -36. Therefore these values do not satisfy the equation or do not belong to the required number set.

Common Pitfalls:
A common mistake is to forget the negative square root and assume only x - 2 = 6, losing one valid root. Another is to overlook the natural number condition and select both positive and negative solutions. Students should always check the domain or set specified in the question and verify solutions against it.

Final Answer:
The only solution that both satisfies the equation and lies in the natural numbers is 8.