Let x be a non-zero number such that the sum of the number and four times its reciprocal is 17/2. Which of the following is a possible value of x?

Introduction / Context:
This problem involves a simple algebraic equation that connects a number with its reciprocal. Such questions test your ability to translate a verbal condition into an algebraic expression and then solve the resulting quadratic equation. Recognizing patterns in expressions that involve a variable and its reciprocal is very useful in many aptitude and algebra questions.

Given Data / Assumptions:

• x is a non-zero real number.
• The relationship given is: x + 4 * (1/x) = 17/2.
• We must determine which option for x satisfies this equation.

Concept / Approach:
To solve equations involving a number and its reciprocal, we multiply through by the variable to eliminate the denominator. This usually leads to a quadratic equation in standard form a*x^2 + b*x + c = 0. We can then solve this quadratic using factorization or the quadratic formula. After finding the roots, we check them against the given options and discard any that are not listed.

Step-by-Step Solution:
Step 1: Start from the equation: x + 4/x = 17/2. Step 2: Multiply both sides by 2x to remove the denominator: 2x * x + 2x * (4/x) = 2x * 17/2. Step 3: Simplify terms: 2x^2 + 8 = 17x. Step 4: Rearrange into standard quadratic form: 2x^2 - 17x + 8 = 0. Step 5: Use the quadratic formula: x = [17 ± sqrt(17^2 - 4 * 2 * 8)] / (2 * 2). Step 6: Compute the discriminant: 17^2 = 289; 4 * 2 * 8 = 64; so discriminant = 289 - 64 = 225. Step 7: The square root of 225 is 15, so x = (17 ± 15) / 4. Step 8: This gives x = (17 + 15) / 4 = 32/4 = 8 or x = (17 - 15) / 4 = 2/4 = 1/2.

Verification / Alternative Check:
Check x = 8 in the original equation: 8 + 4 * (1/8) = 8 + 4/8 = 8 + 1/2 = 8.5 = 17/2, so x = 8 satisfies the condition. The other solution, x = 1/2, also satisfies the equation but it does not appear among the options. Therefore, the only allowable value from the options is 8.

Why Other Options Are Wrong:
If x = 4, then 4 + 4 * (1/4) = 4 + 1 = 5, which is not 17/2.
If x = 12, then 12 + 4 * (1/12) = 12 + 1/3 ≈ 12.33, not 8.5.
If x = 16, then 16 + 4 * (1/16) = 16 + 1/4 = 16.25, again not 17/2. Thus these options do not satisfy the given equation.

Common Pitfalls:
Students sometimes forget to multiply by x correctly and make sign errors when bringing all terms to one side of the equation. Another common mistake is to ignore the second root of the quadratic. Even though 1/2 is a correct mathematical solution, you must still check the answer choices and select the root that is actually listed.

Final Answer:
A value of x that satisfies the condition and appears in the options is 8.