In algebra, let a be a non zero real number satisfying the equation 3a − (3/a) − 3 = 0. Using this condition, evaluate the expression a^3 − (1/a^3) + 2 exactly.

Introduction / Context:
This question links a rational equation in a with an expression involving higher powers of a and its reciprocal. Instead of solving for a numerically and then computing a^3 − 1/a^3, it is more efficient to rearrange the given equation to obtain a simpler relationship, and then use standard identities for cubes and reciprocals. This style of question is common in algebraic manipulation sections of aptitude exams.

Given Data / Assumptions:
- a is a non zero real number, so 1/a is well defined.
- The equation 3a − (3/a) − 3 = 0 holds exactly.
- We are required to find the exact value of a^3 − 1/a^3 + 2.
- Standard algebraic identities and operations on fractions apply.

Concept / Approach:
First, rearrange the given equation to obtain a relation between a and 1/a. Multiplying through by a removes the denominator and gives a quadratic equation in a. Solving this quadratic yields possible values of a, but we do not need to compute them explicitly. Instead, we can use the relation derived from the quadratic to compute a − 1/a and then a^3 − 1/a^3 using the identity a^3 − 1/a^3 = (a − 1/a)(a^2 + a·(1/a) + 1/a^2). Finally, adding 2 gives the target expression.

Step-by-Step Solution:
1) Start from 3a − (3/a) − 3 = 0. 2) Multiply the entire equation by a (which is non zero) to clear the denominator: 3a^2 − 3 − 3a = 0. 3) Rearrange to obtain 3a^2 − 3a − 3 = 0. Divide by 3 to simplify: a^2 − a − 1 = 0. 4) This quadratic shows that a satisfies a^2 = a + 1. 5) From a^2 − a − 1 = 0, we can write 1/a = a − 1 by dividing both sides of a^2 − a − 1 = 0 by a. 6) Therefore, a − 1/a = a − (a − 1) = 1. 7) Use the identity a^3 − 1/a^3 = (a − 1/a)(a^2 + a·(1/a) + 1/a^2). We already have a − 1/a = 1. 8) From a^2 = a + 1 and 1/a = a − 1, we get 1/a^2 = (a − 1)^2 = a^2 − 2a + 1 = (a + 1) − 2a + 1 = 2 − a. 9) Compute a^2 + a·(1/a) + 1/a^2 = (a + 1) + 1 + (2 − a) = a + 1 + 1 + 2 − a = 4. 10) Therefore a^3 − 1/a^3 = (a − 1/a) · 4 = 1 · 4 = 4. 11) Finally, a^3 − 1/a^3 + 2 = 4 + 2 = 6.

Verification / Alternative check:
We can explicitly solve the quadratic a^2 − a − 1 = 0 to find a = (1 ± √5) / 2. Substituting either root into 3a − 3/a − 3 shows that the original equation holds. Then, using a symbolic calculation, we can verify that a^3 − 1/a^3 is 4 for each root, so adding 2 always yields 6. Since both roots give the same value for the target expression, our identity based method is confirmed as correct.

Why Other Options Are Wrong:
Option B (4) is the value of a^3 − 1/a^3 before adding 2. Option C (2) and Option D (0) correspond to incomplete use of identities or mis applying the quadratic relation. Option E (−2) would require a^3 − 1/a^3 to be −4, which contradicts the derived relationship. Only Option A matches the completely simplified value of a^3 − 1/a^3 + 2.

Common Pitfalls:
Some learners try to solve for a numerically and then approximate the expression, which is more work and prone to rounding errors. Others may incorrectly manipulate the equation 3a − 3/a − 3 = 0, for example by dropping the 3 or mis multiplying by a. It is also easy to forget the +2 at the end after finding a^3 − 1/a^3. Keeping track of each transformation and using identities systematically helps avoid mistakes.

Final Answer:
Using the quadratic relation a^2 − a − 1 = 0 and cube identities, we find that a^3 − 1/a^3 + 2 = 6.