If a is a non-zero real number satisfying the relation a − (1/a) = 7, find the value of: a^2 + (1/a^2) Choose the correct value.

Introduction / Context: This problem checks algebraic manipulation of expressions involving a and 1/a. The key idea is to square a given relation and rewrite it in terms of a^2 + 1/a^2.

Given Data / Assumptions:

• a − (1/a) = 7 • a is real and a ≠ 0 • Required: a^2 + (1/a^2)

Concept / Approach: Use the identity: (a − 1/a)^2 = a^2 − 2 + 1/a^2 So if we know (a − 1/a), squaring it directly gives a^2 + 1/a^2 after adding 2 on both sides.

Step-by-Step Solution: 1) Start with: a − (1/a) = 7 2) Square both sides: (a − 1/a)^2 = 7^2 = 49 3) Expand: (a − 1/a)^2 = a^2 − 2*(a*(1/a)) + 1/a^2 4) Since a*(1/a) = 1, we get: a^2 − 2 + 1/a^2 5) So: a^2 − 2 + 1/a^2 = 49 6) Add 2 to both sides: a^2 + 1/a^2 = 51

Verification / Alternative check: If (a − 1/a)^2 is 49, then a^2 + 1/a^2 must be 49 + 2 = 51 by the identity. This is consistent and does not require finding a itself.

Why Other Options Are Wrong: • 49: forgets the +2 adjustment from the identity. • 52, 53, 54: would imply (a − 1/a)^2 is 50, 51, 52, which contradicts 49.

Common Pitfalls: • Expanding (a − 1/a)^2 incorrectly. • Forgetting that a*(1/a) = 1, not a.

Final Answer: 51