If 2x + 2/x = 3, then the value of x3 + 1/x3 + 2 is

Concept: For any nonzero x, the identity (x + 1/x)3 = x3 + 1/x3 + 3(x + 1/x) connects the sum x + 1/x to the sum of cubes x3 + 1/x3. Once x + 1/x is known, x3 + 1/x3 follows directly from this identity, without ever needing to solve for x itself.

Application -- step by step:

1. Divide the given equation 2x + 2/x = 3 by 2 to get x + 1/x = 3/2.

2. Cube both sides using the identity: (x + 1/x)3 = x3 + 1/x3 + 3(x + 1/x), so (3/2)3 = x3 + 1/x3 + 3(3/2).

3. Compute the two known quantities: (3/2)3 = 27/8, and 3 × 3/2 = 9/2 = 36/8.

4. Solve for the unknown sum: x3 + 1/x3 = 27/8 - 36/8 = -9/8.

5. Add the constant the question asks for: x3 + 1/x3 + 2 = -9/8 + 16/8 = 7/8.

Cross-check: x3 + 1/x3 also factors as (x + 1/x)(x2 - x·(1/x) + 1/x2). Since x·(1/x) = 1 for any nonzero x, and x2 + 1/x2 = (x + 1/x)2 - 2 = 9/4 - 2 = 1/4, the bracket becomes 1/4 - 1 = -3/4. Multiplying by x + 1/x = 3/2 gives (3/2)(-3/4) = -9/8, the same value found in step 4, confirming the result.

Result: So x3 + 1/x3 + 2 = 7/8.