Given that x + 1/(4x) = 5/2 for a non zero real number x, what is the exact value of the expression (64x^6 + 1) / (8x^3), simplified as a single real number?

Introduction / Context:
This question checks your ability to use algebraic identities and clever substitutions rather than solving directly for x. The given relation x + 1/(4x) = 5/2 helps transform a higher power expression into something simpler. Recognising patterns such as u + 1/u and using them correctly is a common technique in aptitude level algebra.

Given Data / Assumptions:

• We know x + 1/(4x) = 5/2.
• x is a non zero real number.
• We must evaluate E = (64x^6 + 1) / (8x^3).
• The answer should be an exact numerical value.

Concept / Approach:
Notice that 64x^6 is (8x^3)^2, so the expression (64x^6 + 1) / (8x^3) can be rewritten in the form (u^2 + 1)/u where u = 8x^3. That simplifies to u + 1/u. The given relation involves x and 1/(4x). We can manipulate this relation to obtain x^3 + 1/(64x^3), which is directly related to u + 1/u, because u = 8x^3 and 1/u = 1/(8x^3).

Step-by-Step Solution:
Let a = x + 1/(4x). We are given a = 5/2.Compute a^2: a^2 = x^2 + 1/(16x^2) + 2 * x * 1/(4x) = x^2 + 1/(16x^2) + 1/2.So x^2 + 1/(16x^2) = a^2 − 1/2 = 25/4 − 1/2 = 25/4 − 2/4 = 23/4.Now consider the product (x^2 + 1/(16x^2)) * (x + 1/(4x)) = x^3 + 1/(64x^3) + x/4 + 1/(16x).Notice x/4 + 1/(16x) = (1/4)(x + 1/(4x)) = a/4. Hence the product equals x^3 + 1/(64x^3) + a/4.So x^3 + 1/(64x^3) = a(x^2 + 1/(16x^2)) − a/4 = a(23/4) − a/4 = a * (22/4) = (11/2) * (5/2) = 55/4.Set u = 8x^3. Then x^3 + 1/(64x^3) = (1/8)(u + 1/u). Thus (1/8)(u + 1/u) = 55/4, so u + 1/u = 8 * 55/4 = 110.Finally, E = (64x^6 + 1) / (8x^3) = (u^2 + 1)/u = u + 1/u = 110.

Verification / Alternative check:
If you solve x + 1/(4x) = 5/2 explicitly for x, you will obtain two possible real values. Substituting those values back into E using a calculator will show that E is 110 in both cases. This confirms that the algebraic manipulations and the identity based approach are correct and independent of which valid x is chosen.

Why Other Options Are Wrong:

• 115, 125, 140, and 150 correspond to rough or incorrect approximations that arise when identities are misused or some steps are omitted.
• Only 110 matches the exact expression u + 1/u derived from the given relation.

Common Pitfalls:

• Not spotting that 64x^6 is (8x^3)^2 and missing the chance to rewrite the expression as u + 1/u.
• Squaring a = x + 1/(4x) incorrectly or forgetting the middle term 1/2.
• Errors while organising terms to get x^3 + 1/(64x^3) and then relating it to u + 1/u.

Final Answer:
110