Difficulty: Medium
Correct Answer: 52
Explanation:
Introduction / Context:
This problem is a classic application of algebraic identities involving cubes and reciprocals. Rather than solving for x directly, you are encouraged to work with the simpler expression x − 3, treat it as a new variable, and then use a standard identity to jump from a linear equation to a cubic expression. Such techniques are extremely useful in competitive examinations for saving time and avoiding messy algebra.
Given Data / Assumptions:
Concept / Approach:
The key idea is to let t = x − 3. The given equation becomes t + 1 / t = 4. From this, you can use the identity (t + 1 / t)^3 = t^3 + 1 / t^3 + 3(t + 1 / t). This allows you to compute t^3 + 1 / t^3 directly from the known value of t + 1 / t, without ever finding t itself.
Step-by-Step Solution:
Verification / Alternative check:
If you want to check numerically, solve t + 1 / t = 4 by multiplying through by t: t^2 + 1 = 4t, so t^2 − 4t + 1 = 0. This is a quadratic equation whose roots can be found using the formula, and substituting those roots into t^3 + 1 / t^3 will give the same result 52, confirming the identity based solution.
Why Other Options Are Wrong:
The values 14 and 18 are far too small and usually arise from misusing the identity or forgetting the factor 3(t + 1 / t). The values 64 and 76 come from computing 4^3 alone or adding instead of subtracting 12. Only 52 is consistent with the correct algebraic manipulation.
Common Pitfalls:
Learners sometimes attempt to solve directly for x, which involves additional algebraic steps and increases the risk of errors. Another frequent mistake is to misremember the cube identity and write t^3 + 1 / t^3 = (t + 1 / t)^3 − 3, omitting the factor (t + 1 / t) inside the correction term. Carefully recalling the full identity prevents such errors.
Final Answer:
The exact value of (x − 3)^3 + 1 / (x − 3)^3 is 52.
Discussion & Comments