For a non-zero real number x, expressions like x + 1/x can be used to find higher powers without solving for x directly.\nIf x + 1/x = 6, what is the value of x^2 + 1/x^2?

Introduction / Context:
This problem tests an algebraic identity technique: using (x + 1/x)^2 to quickly obtain x^2 + 1/x^2. This avoids solving a quadratic for x and is a common simplification shortcut in aptitude and algebra questions.

Given Data / Assumptions:

• x is a non-zero real number.
• x + 1/x = 6.
• We must find x^2 + 1/x^2.

Concept / Approach:
Square the given expression. The identity is:\n(x + 1/x)^2 = x^2 + 2 + 1/x^2.\nThen rearrange to isolate x^2 + 1/x^2. This works because the cross term is always 2 when multiplying x by 1/x.

Step-by-Step Solution:
Given: x + 1/x = 6 Square both sides: (x + 1/x)^2 = 6^2 Left side expansion: (x + 1/x)^2 = x^2 + 2*(x*(1/x)) + 1/x^2 Since x*(1/x) = 1, the middle term becomes 2 So: x^2 + 2 + 1/x^2 = 36 Rearrange: x^2 + 1/x^2 = 36 - 2 = 34

Verification / Alternative check:
If desired, you could solve x + 1/x = 6 => x^2 - 6x + 1 = 0 and use the solutions. Both solutions will produce the same value for x^2 + 1/x^2, confirming the identity-based result.

Why Other Options Are Wrong:
36 is the value of (x + 1/x)^2, not x^2 + 1/x^2.
32 is a common mistake from subtracting 4 instead of 2.
23 and 16 do not match the correct rearrangement and are inconsistent with the squared identity.

Common Pitfalls:
Forgetting the cross term 2, or incorrectly writing (x + 1/x)^2 = x^2 + 1/x^2. Another mistake is treating x*(1/x) as x rather than 1.

Final Answer:
x^2 + 1/x^2 = 34