If x + 1/x = 5 for a non zero real number x, then what is the value of the rational expression 5x / (x^2 + 5x + 1)?

Introduction / Context:
This question is a typical example of using the given value of x + 1/x to simplify a more complicated rational expression involving x. Instead of solving for x explicitly, we convert the denominator into a form that can be expressed using x + 1/x and then simplify the expression directly.

Given Data / Assumptions:

• x is a non zero real number.
• x + 1/x = 5.
• We must find 5x / (x^2 + 5x + 1).

Concept / Approach:
From x + 1/x = 5 we can derive the quadratic equation satisfied by x. Then we use that equation to rewrite x^2 + 5x + 1 in terms of x. Once the denominator is simplified, the fraction becomes much easier to evaluate without solving for x numerically.

Step-by-Step Solution:
Step 1: Start from x + 1/x = 5. Step 2: Multiply both sides by x to clear the denominator: x^2 + 1 = 5x. Step 3: Rearrange to form a quadratic: x^2 - 5x + 1 = 0. Step 4: Now consider the denominator x^2 + 5x + 1. Step 5: Replace x^2 using x^2 = 5x - 1 from the quadratic. Step 6: Then x^2 + 5x + 1 = (5x - 1) + 5x + 1 = 10x. Step 7: Substitute this into the expression: 5x / (x^2 + 5x + 1) = 5x / 10x. Step 8: Cancel x (x is non zero) to get 5/10 = 1/2.

Verification / Alternative check:
We can solve the quadratic x^2 - 5x + 1 = 0 to find the actual roots, but whichever root we choose, substituting back into 5x / (x^2 + 5x + 1) yields 1/2. Since the expression simplifies algebraically to a constant, it is independent of the specific root, which confirms our result.

Why Other Options Are Wrong:
Options A (1/3), B (1/4), and D (1/5) would result from incorrect manipulation, such as miscomputing x^2 or failing to simplify x^2 + 5x + 1 correctly. Option E (2/3) could appear if a student incorrectly cancels terms or does not handle the factor 10x properly in the denominator.

Common Pitfalls:
A common error is to square x + 1/x unnecessarily or to attempt to solve for x with approximate decimals. Another pitfall is cancelling incorrectly before fully simplifying the denominator. It is safer to express x^2 from the quadratic and substitute carefully.

Final Answer:

The value of the expression is 1/2.