For real numbers x, y, and z, suppose: (2x − y)^2 + (3y − 2z)^2 = 0 What is the ratio x : y : z? Choose the correct ratio.

Introduction / Context:
This question uses a key property of real numbers: a sum of squares can be zero only when each squared term is zero. That converts the problem into a simple system of linear equations and then a ratio.

Given Data / Assumptions:

• (2x − y)^2 + (3y − 2z)^2 = 0 • x, y, z are real numbers • Required: ratio x : y : z

Concept / Approach:
For real numbers, each square is always ≥ 0. The only way their sum equals 0 is if both squares are exactly 0: (2x − y)^2 = 0 and (3y − 2z)^2 = 0. Then solve the resulting linear equations and convert to a ratio.

Step-by-Step Solution:
1) Since (2x − y)^2 ≥ 0 and (3y − 2z)^2 ≥ 0, and their sum is 0, each must be 0. 2) Set first term to zero: (2x − y)^2 = 0 → 2x − y = 0 → y = 2x 3) Set second term to zero: (3y − 2z)^2 = 0 → 3y − 2z = 0 → 2z = 3y → z = 3y/2 4) Substitute y = 2x into z: z = 3*(2x)/2 = 3x 5) Therefore: x : y : z = x : 2x : 3x = 1 : 2 : 3

Verification / Alternative check:
Take x = 1, y = 2, z = 3. Then: 2x − y = 2 − 2 = 0 and 3y − 2z = 6 − 6 = 0, so the sum of squares is 0, confirming the ratio works.

Why Other Options Are Wrong:
• Any ratio not satisfying y = 2x and z = 3x will make at least one squared term positive, so the sum cannot be 0.

Common Pitfalls:
• Thinking the terms can cancel each other (they cannot because squares are non-negative). • Forgetting to take square-root logic properly: if t^2 = 0, then t = 0 only.

Final Answer:
1 : 2 : 3