If (x / 5) + (5 / x) = −2, find the exact value of x^3.

Introduction / Context:

This question focuses on solving a rational equation and then using the result to find a power of x. Rather than dealing directly with x, it is often easier to introduce a substitution that simplifies the fraction. This is a common technique in aptitude questions, where a clever substitution can turn a complicated expression into a simple quadratic equation.

Given Data / Assumptions:

• The equation (x / 5) + (5 / x) = −2 holds.
• x is nonzero so that the term 5 / x is defined.
• You are required to find x^3.

Concept / Approach:

A natural substitution is t = x / 5. This turns the reciprocal term 5 / x into a simple 1 / t. The equation then becomes t + 1 / t = −2, which is a standard form. Multiplying through by t gives a quadratic equation in t. Once you solve for t, you can recover x and then compute x^3 directly.

Step-by-Step Solution:

Verification / Alternative check:

Check that x = −5 satisfies the original equation. Substitute x = −5: (x / 5) + (5 / x) = (−5 / 5) + (5 / −5) = −1 + (−1) = −2, which matches the given equation exactly. Then x^3 = −125, confirming the solution is consistent.

Why Other Options Are Wrong:

The value 5 would correspond to x = cube root of 5, which does not satisfy the initial equation. The value 1/125 comes from mistakenly using x = 1/5. The value 625 corresponds to x = 5, which makes (x / 5) + (5 / x) equal to 2, not −2. The value 125 would require x = 5, again failing the original condition. Only −125 matches both the equation and the derived value of x^3.

Common Pitfalls:

Some learners attempt to clear denominators by multiplying directly by x, which leads to more complicated algebra. Others forget to consider that t could be negative and make errors in factoring the quadratic. Using a simple substitution and carefully solving the quadratic keeps the solution clean.

Final Answer:

The value of x^3 is −125.