If the expression (x − 2)^2 + (y + 3)^2 + (z − 15)^2 is equal to 0 for real numbers x, y, and z, what is the value of x + y + z − 5?

Introduction / Context:
This algebraic question uses the property that a sum of squares of real numbers can only be zero if each squared term itself is zero. It is a straightforward but important idea that often appears in coordinate geometry and optimisation problems.

Given Data / Assumptions:

• (x − 2)^2 + (y + 3)^2 + (z − 15)^2 = 0.
• x, y, and z are real numbers.
• We must compute x + y + z − 5.

Concept / Approach:
For any real number t, t^2 is always greater than or equal to zero, and it equals zero only when t = 0. Therefore, a sum of several squared terms can equal zero only if every individual squared term is zero. We apply this reasoning to each bracketed expression, solve for x, y, and z, and then substitute into x + y + z − 5.

Step-by-Step Solution:
We have (x − 2)^2 + (y + 3)^2 + (z − 15)^2 = 0. Each term is a square of a real number, so each is at least 0. The sum of three non negative numbers is zero only if all three are zero individually. Thus (x − 2)^2 = 0, (y + 3)^2 = 0, and (z − 15)^2 = 0. From (x − 2)^2 = 0 we get x − 2 = 0, so x = 2. From (y + 3)^2 = 0 we get y + 3 = 0, so y = −3. From (z − 15)^2 = 0 we get z − 15 = 0, so z = 15. Now compute x + y + z − 5 = 2 + (−3) + 15 − 5. Simplify the sum: 2 − 3 = −1, and −1 + 15 = 14, then 14 − 5 = 9.
Verification / Alternative check:
Substitute x = 2, y = −3, and z = 15 back into the original expression. Each square becomes zero: (2 − 2)^2 = 0, (−3 + 3)^2 = 0, and (15 − 15)^2 = 0, so the total sum is 0 as required. This confirms that the values of x, y, and z are consistent with the given condition.

Why Other Options Are Wrong:
Options a, c, d, and e correspond to different values of x + y + z − 5 that do not follow from the only possible triple (2, −3, 15) that makes the sum of squares zero. Since there is a unique combination of x, y, and z that satisfies the equation, the resulting expression x + y + z − 5 is uniquely determined as 9.

Common Pitfalls:
Some learners may overlook the fact that each squared term must be zero individually and try to solve the equation in a more complicated way. Others may make arithmetic errors when computing the final sum. Remembering the basic property of squares and checking arithmetic carefully is sufficient here.

Final Answer:
The value of x + y + z − 5 is 9.