When x + 1 / x = 5, find the value of x − 1 / x.

Introduction / Context:

This problem illustrates how to move between x + 1 / x and x − 1 / x using algebraic identities. You are given the sum of x and its reciprocal and asked to find their difference. This type of question appears often in algebra sections of aptitude exams and rewards a clear understanding of squared expressions and how they relate to each other.

Given Data / Assumptions:

• x + 1 / x = 5.
• x is nonzero so the reciprocal 1 / x is defined.
• You must find x − 1 / x.
• x may be positive or negative, so both signs are considered where appropriate.

Concept / Approach:

The key tool is to square x + 1 / x and relate it to x^2 + 1 / x^2. A similar relation holds for x − 1 / x. By first finding x^2 + 1 / x^2 from the given sum and then using another identity for (x − 1 / x)^2, you can solve for x − 1 / x up to a sign. Because the original equation does not fix the sign of x, the answer is naturally a plus or minus value.

Step-by-Step Solution:

Verification / Alternative check:

You can verify by solving the original equation for x. Multiply x + 1 / x = 5 by x to get x^2 − 5x + 1 = 0. This quadratic has two real roots, one greater than 1 and one between 0 and 1. For each root, compute x − 1 / x numerically; you will obtain the positive and negative values of √21, confirming the plus and minus form of the answer.

Why Other Options Are Wrong:

The numbers 11 and 21 would only occur if one mistakenly added instead of subtracting in the squared expressions or ignored the identities. The choice ± √22 comes from miscalculating x^2 + 1 / x^2 as 24 instead of 23. The value 0 would require x = 1 / x, that is x^2 = 1, which does not satisfy x + 1 / x = 5. Only ± √21 is consistent with the algebraic relations derived.

Common Pitfalls:

Students often forget the middle term when squaring x + 1 / x, writing x^2 + 1 / x^2 instead of x^2 + 2 + 1 / x^2. Another common error is to forget that taking the square root yields both positive and negative results, not just one sign. Being careful with these details is crucial for correct answers in exam settings.

Final Answer:

The value of x − 1 / x, given x + 1 / x = 5, is ± √21.