Solve the equation 12^(2x + 1) = 1 and find the real value of x.

Introduction / Context:
This question is a simple exponential equation that checks whether students understand how to solve equations of the form a^(f(x)) = 1. It also connects the behaviour of exponential functions with their exponents. Such problems are common in aptitude and entrance tests because they combine algebra with basic properties of exponents and do not require heavy computation.

Given Data / Assumptions:
- The equation is 12^(2x + 1) = 1.
- The base 12 is a positive real number and not equal to 1.
- We are looking for real solutions for x.

Concept / Approach:
For an exponential expression of the form a^k with a greater than 0 and a not equal to 1, the value a^k equals 1 if and only if the exponent k equals 0. This is because a^0 = 1 for any such base a, and there is no other real exponent that gives 1 when the base is not 1. Therefore we can solve this equation simply by setting the exponent 2x + 1 equal to 0 and solving the resulting linear equation.

Step-by-Step Solution:
We are given 12^(2x + 1) = 1. Note that 12 is greater than 0 and not equal to 1. For such a base, a^k = 1 implies k = 0. Therefore set the exponent equal to zero: 2x + 1 = 0. Solve the linear equation: 2x = -1. Then x = -1 / 2. Thus the unique real solution of the original equation is x = -1 / 2.

Verification / Alternative check:
Substitute x = -1 / 2 back into the original equation. The exponent becomes 2(-1 / 2) + 1 = -1 + 1 = 0. Therefore the left hand side is 12^0 = 1. This matches the right hand side exactly, confirming that x = -1 / 2 is correct. Because the exponential function 12^(2x + 1) is continuous and strictly monotonic in x, there can be no other real solution to this equation.

Why Other Options Are Wrong:
1 / 2: Substituting x = 1 / 2 gives exponent 2(1 / 2) + 1 = 2, so 12^2 which is 144, not 1.
1: Substituting x = 1 gives exponent 2(1) + 1 = 3, so 12^3 which is 1728, not 1.
-1: Substituting x = -1 gives exponent 2(-1) + 1 = -1, so 12^-1 which is 1 / 12, not 1.

Common Pitfalls:
Some students mistakenly think that any exponent can give 1 if the base is large, which is not correct. Others confuse this situation with a^k = 0 or a^k = a and try to set the exponent to 1 instead of 0. Another error is to attempt to take logarithms on both sides without realizing that the simple property a^0 = 1 makes the problem trivial. Remember that for a base a not equal to 1, the only way to get a^k = 1 is to have k equal to 0.

Final Answer:
The solution of the equation 12^(2x + 1) = 1 is x = -1 / 2.