Basic Mathematics & Logarithm
Equations — Sum of Squares
Grade Class 11
Question:
<p>If \(4x^2 + 4y^2 + 4z^2 - 2y - 2yz - zx = 0\), then \(x : y : z =\)</p>
1 : 2 : 1
2 : 1 : 2
1 : 2 : 3
1 : 1 : 2
Step-by-Step Solution
Key Concept: Rearrange as a sum of squared binomials equal to zero. Equate each squared term to zero.
Notice that the best first move is to reveal the hidden structure in the expression. A clever move here is to rewrite the problem in the form where the standard theorem or identity applies cleanly. Rewrite: $(2x - z/2)^2 + (2y - z/2)^2 + (2z - \text{...})^2 = 0$. Each term must be zero. Solving the system gives $x:y:z = 2:1:2$. Now, we invoke the power of that idea, simplify patiently, and then check that the final answer really fits the original problem.
Correct Answer: B