If x + 1/x = 5 for a nonzero real number x, what is the value of the expression 6x / (x² + x + 1)?

Introduction / Context:
This algebra question uses a relation between x and 1/x to simplify another rational expression in x. Instead of solving a quadratic equation for x, we can manipulate the given identity to rewrite the denominator, leading to a quick simplification. These techniques are very useful in aptitude exams to save time.

Given Data / Assumptions:

• x + 1/x = 5.
• x ≠ 0, since 1/x exists.
• We need to compute 6x / (x² + x + 1).

Concept / Approach:
Main ideas:

• From x + 1/x = 5, derive a relationship involving x² and 1.
• Try to express the denominator x² + x + 1 in terms of x and the known relation.
• Once the denominator is expressed as a multiple of x, cancel with the numerator 6x to simplify.

Step-by-Step Solution:
Step 1: Start with the given equation: x + 1/x = 5. Step 2: Multiply both sides by x to clear the fraction: x² + 1 = 5x. Step 3: Rearrange to get x² − 5x + 1 = 0, which is a quadratic that x satisfies. Step 4: We want to simplify x² + x + 1. From x² + 1 = 5x, add x to both sides to get x² + x + 1 = 5x + x = 6x. Step 5: Substitute this into the expression: 6x / (x² + x + 1) = 6x / (6x). Step 6: Since x ≠ 0, 6x cancels, giving 6x / 6x = 1.
Verification / Alternative check:
Step 1: Solve the quadratic x² − 5x + 1 = 0 using the quadratic formula if desired. Step 2: For each root obtained, compute x + 1/x and confirm it equals 5. Step 3: Substitute those numeric values of x back into 6x / (x² + x + 1) and verify that the result is 1 in both cases, confirming that the expression is constant for the given condition.
Why Other Options Are Wrong:
Options 3 and 2 would result only if the denominator were 2x or 3x rather than 6x. Option 0 would require the numerator to be zero, which is not true since x ≠ 0. Option 5 is simply the value of x + 1/x and has no direct connection to the simplified expression after substitution.
Common Pitfalls:
Some students incorrectly treat x² + x + 1 as x(x + 1) + 1 and then attempt unrelated factoring rather than using the relation x² + 1 = 5x. Another mistake is to solve for x immediately, which is not necessary and consumes more time. Careless algebraic manipulation when adding x to both sides can also lead to errors such as writing x² + x + 1 = 5x − x.
Final Answer:
The value of 6x / (x² + x + 1) is 1.