If a number and its reciprocal have a sum of 10/3, determine the number(s).

Difficulty: Easy

3, 1/3
3, -1/3
-3, 1/3
-3, -1/3
10/3 only

Correct Answer: 3, 1/3

Explanation:

Introduction / Context:
Sum-with-reciprocal problems lead to a quadratic after clearing denominators. Solving that quadratic yields the pair of numbers, which are mutual reciprocals by construction.

Given Data / Assumptions:

x + 1/x = 10/3.
x ≠ 0.

Concept / Approach:
Multiply by x to obtain a standard quadratic in x. Solve using the quadratic formula or factoring if the discriminant is a perfect square.

Step-by-Step Solution:

x + 1/x = 10/3 ⇒ multiply both sides by x: x^2 + 1 = (10/3)x.Rearrange: 3x^2 − 10x + 3 = 0.Discriminant Δ = (−10)^2 − 4*3*3 = 100 − 36 = 64.Roots: x = (10 ± 8)/6 ⇒ 18/6 = 3 or 2/6 = 1/3.

Verification / Alternative check:
Check: 3 + 1/3 = 10/3; also 1/3 + 3 = 10/3, both valid.

Why Other Options Are Wrong:
Options with negative signs do not satisfy the given positive sum; “10/3 only” is not a number satisfying x + 1/x = 10/3.

Common Pitfalls:
Forgetting to multiply through by x or mishandling the quadratic coefficients.

If a number and its reciprocal have a sum of 10/3, determine the number(s).

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