If 2x + 1/(3x) = 5 for a non zero real number x, then what is the value of 5x / (6x^2 + 20x + 1)?

Introduction / Context:
This problem is another example of simplifying a rational expression using a given equation that relates x and 1/x. The goal is to avoid solving for x directly and instead use algebraic manipulation to transform the denominator into something proportional to x, which then simplifies nicely.

Given Data / Assumptions:

• x is a non zero real number.
• 2x + 1/(3x) = 5.
• We must find 5x / (6x^2 + 20x + 1).

Concept / Approach:
We begin by clearing the denominator in the given equation to obtain a quadratic equation in x. This equation lets us express 1 in terms of x and then rewrite 6x^2 + 20x + 1 in a form involving x(6x + 20) plus 1. Manipulating the quadratic carefully, we can reduce the entire fraction.

Step-by-Step Solution:
Step 1: Start from 2x + 1/(3x) = 5. Step 2: Multiply both sides by 3x (x is non zero): 3x(2x) + 3x(1/(3x)) = 5(3x). Step 3: This simplifies to 6x^2 + 1 = 15x, so 6x^2 - 15x + 1 = 0. Step 4: Rearrange to express 1 in terms of x: 1 = 15x - 6x^2. Step 5: Consider the denominator D = 6x^2 + 20x + 1. Step 6: Substitute 1 = 15x - 6x^2 into D: D = 6x^2 + 20x + (15x - 6x^2) = 35x. Step 7: Now compute the expression: 5x / D = 5x / 35x = 5/35 = 1/7 (since x is not zero, we can cancel x).

Verification / Alternative check:
We can solve the quadratic 6x^2 - 15x + 1 = 0 to obtain two possible values for x, and for each solution, direct substitution into 5x / (6x^2 + 20x + 1) yields 1/7. This confirms that the simplification is correct and independent of the particular root chosen.

Why Other Options Are Wrong:
Options A (1/4), B (1/6), C (1/5), and E (1/3) result from incorrect rearrangement of 6x^2 - 15x + 1 = 0 or from errors when substituting into the denominator. For instance, forgetting the sign when moving terms or miscalculating the combined coefficient of x in the denominator will lead to a wrong constant.

Common Pitfalls:
Learners often make sign mistakes when moving 15x to the left side or misapply the idea of expressing 1 in terms of x. Another pitfall is failing to notice that the denominator simplifies to a multiple of x, which makes the final cancellation very simple.

Final Answer:

The value of the expression is 1/7.