If the sum of a non-zero real number and its reciprocal is 2, what is the value of the number?

Introduction / Context:
This is a classic algebra question involving a number and its reciprocal. You are told that the sum of a non-zero real number and its reciprocal equals 2, and you must determine the number. The problem leads naturally to a quadratic equation and tests basic equation solving skills and understanding of reciprocals.

Given Data / Assumptions:

- Let the non-zero real number be x. - Its reciprocal is 1 / x. - The equation given is x + 1 / x = 2. - x cannot be 0 because the reciprocal 1 / x would be undefined.

Concept / Approach:
The usual method is to clear the denominator by multiplying the equation by x, which results in a quadratic equation in x. We then solve this quadratic, either by factoring or using the quadratic formula. It is important to remember domain restrictions: x must be non zero. After finding all algebraic solutions, we check them against the options and the domain condition before choosing the final answer.

Step-by-Step Solution:
Step 1: Start from the equation x + 1 / x = 2. Step 2: Multiply both sides of the equation by x to remove the fraction: x^2 + 1 = 2x. Step 3: Rearrange to standard quadratic form: x^2 - 2x + 1 = 0. Step 4: Recognize that x^2 - 2x + 1 is a perfect square: x^2 - 2x + 1 = (x - 1)^2. Step 5: So the equation becomes (x - 1)^2 = 0. Step 6: Take the square root of both sides: x - 1 = 0. Step 7: Hence x = 1. Step 8: There is only one real solution, and it satisfies x not equal to 0.

Verification / Alternative check:
Substitute x = 1 back into the original equation to verify: x + 1 / x = 1 + 1 / 1 = 1 + 1 = 2. This matches the given sum, so x = 1 is indeed a valid solution. Since the quadratic factors as a perfect square, there are no other distinct real solutions to check, and no extraneous roots are introduced by clearing the denominator as x is non zero.

Why Other Options Are Wrong:
Option A (0) is invalid because the reciprocal 1 / 0 is undefined. Option C (-1) gives -1 + (-1) = -2, not 2. Option D (2) gives 2 + 1/2 = 2.5, which is also not equal to 2. Only x = 1 satisfies the given equation and respects the non-zero condition on the number.

Common Pitfalls:
Some learners may try to guess the answer and overlook the non-zero restriction, or they might attempt to square both sides directly, which is unnecessary and can create extraneous solutions. Others might incorrectly factor the quadratic or mis-handle signs when rearranging terms. Writing the equation carefully, using standard algebraic techniques and then verifying by substitution helps avoid these errors.

Final Answer:
The number whose sum with its reciprocal is 2 is 1, which corresponds to option B.