In aptitude (algebra and powers), consider the quadratic equation x^2 - 2x + 1 = 0. First solve this equation for its real root x, and then, using this value, compute the exact value of the expression x^4 + 1/x^4.

Introduction / Context:
This aptitude question combines a very basic quadratic equation with manipulation of powers and reciprocals. The key idea is that once we know the exact value of x from the quadratic, the expression x^4 + 1/x^4 becomes a simple substitution problem. Such questions are common in school algebra and competitive exams because they test both equation solving and comfort with indices in a single step.

Given Data / Assumptions:
- We are given the quadratic equation x^2 - 2x + 1 = 0.
- We are asked to find x^4 + 1/x^4 once x is known.
- x must be non zero, otherwise 1/x^4 would not be defined. The quadratic itself will confirm this.

Concept / Approach:
First, solve the quadratic equation using factoring. Then, once the root is known, substitute this root into the expression x^4 + 1/x^4. Because the quadratic has a repeated root, we only get one possible real value of x, which makes the resulting expression unique. The arithmetic in powers is very simple once x is found.

Step-by-Step Solution:
1) Rewrite the quadratic: x^2 - 2x + 1 = 0. 2) Factor it as (x - 1)(x - 1) = 0, which means x - 1 = 0. 3) Therefore x = 1 is the only real root, and it is non zero, so 1/x^4 is defined. 4) Compute x^4 = 1^4 = 1. 5) Compute 1/x^4 = 1/1^4 = 1. 6) Add the two values: x^4 + 1/x^4 = 1 + 1 = 2.

Verification / Alternative check:
We can quickly verify the factorisation by expanding (x - 1)^2. This gives x^2 - 2x + 1, which matches the original quadratic exactly. Also, substituting x = 1 directly into the original equation gives 1 - 2 + 1 = 0, confirming that x = 1 is correct. Once that is confirmed, the values 1^4 and 1/1^4 are clearly equal to 1, so the sum 2 is secure.

Why Other Options Are Wrong:
Option A (1) would be correct if the expression were x^4 or 1/x^4 alone, but not their sum. Option C (0) would require the two terms to cancel, which would need x^4 = -1/x^4, impossible for real x. Option D (3) and Option E (4) are simple distractors obtained by guessing or by incorrectly adding extra terms, but none matches the exact computed sum of 2.

Common Pitfalls:
A common mistake is to mis factor the quadratic or to assume two different roots instead of recognising it as a perfect square. Another error is to forget that both x^4 and 1/x^4 must be evaluated at the same root. Some learners also rush and write x^4 + 1/x^4 = (x + 1/x)^4, which is algebraically incorrect. Careful stepwise substitution avoids all of these issues.

Final Answer:
The quadratic equation has root x = 1, and substituting this value into the expression gives x^4 + 1/x^4 = 2, so the correct answer is 2.