If real numbers x, y and z satisfy (x − 3)^2 + (y − 4)^2 + (z − 5)^2 = 0, then what is the value of the sum x + y + z?

Introduction / Context:
This is a geometry flavored algebra question involving squares of differences. A sum of squares equal to zero has a very strong implication about each term, which we use to find x, y, and z and then compute x + y + z.

Given Data / Assumptions:

• (x − 3)^2 + (y − 4)^2 + (z − 5)^2 = 0.
• x, y, and z are real numbers.
• We must find x + y + z.

Concept / Approach:
For real numbers, a square of any real quantity is always greater than or equal to zero. A sum of several squares can only be zero if each individual square is zero. This principle allows us to determine x, y, and z exactly, without solving any complicated equations.

Step-by-Step Solution:
Step 1: Note that (x − 3)^2 ≥ 0, (y − 4)^2 ≥ 0, and (z − 5)^2 ≥ 0 for all real x, y, z. Step 2: Their sum is given to be zero: (x − 3)^2 + (y − 4)^2 + (z − 5)^2 = 0. Step 3: A sum of non negative numbers can be zero only if each term is zero. Step 4: Therefore (x − 3)^2 = 0, (y − 4)^2 = 0, and (z − 5)^2 = 0. Step 5: This implies x − 3 = 0, y − 4 = 0, and z − 5 = 0. Step 6: Hence x = 3, y = 4, and z = 5. Step 7: The required sum is x + y + z = 3 + 4 + 5 = 12.

Verification / Alternative check:
Substitute x = 3, y = 4, z = 5 back into the original expression: (3 − 3)^2 + (4 − 4)^2 + (5 − 5)^2 = 0 + 0 + 0 = 0, which is consistent with the given condition. Any other value for x, y, or z would make at least one square positive and thus the sum positive, contradicting the equation.

Why Other Options Are Wrong:
Options A (-12), B (0), C (8), and E (4) do not correspond to the unique triple (3, 4, 5) forced by the sum of squares being zero. There is no freedom to choose alternative values because the equation pins down x, y, and z exactly.

Common Pitfalls:
Some learners overlook the fact that each square must itself be zero and instead attempt to expand and solve three variable equations, which is unnecessary. Others may misinterpret the expression as a distance formula but still forget the strict non negativity of squares. Always remember that a sum of real squares is zero only when every term is zero.

Final Answer:

The value of x + y + z is 12.