Question
\( ( x - 2 ) ^ { 2 } + ( 2 y + 5 ) ^ { 2 } = 0 \)
Ask by Mcfarlane Carter. in South Africa
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solution is \( x = 2 \) and \( y = -\frac{5}{2} \).
Solution
The equation
(x - 2)² + (2y + 5)² = 0
involves a sum of two squared expressions. Since the square of any real number is always nonnegative, the only way for their sum to equal zero is for each individual square to be zero. That is, both
(x - 2)² = 0 and (2y + 5)² = 0
must hold simultaneously.
Solving each equation:
1. For (x - 2)² = 0:
x - 2 = 0 ⟹ x = 2
2. For (2y + 5)² = 0:
2y + 5 = 0 ⟹ 2y = -5 ⟹ y = -5/2
Thus, the only solution to the equation is the point (2, -5/2).
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Simplify this solution The Deep Dive
This equation can only hold true when both components equal zero. For \( ( x - 2 ) ^ { 2 } = 0 \), we find \( x = 2 \). Then for \( ( 2 y + 5 ) ^ { 2 } = 0 \), we solve for \( y \) to get \( y = -\frac{5}{2} \). So, the only solution is the point \( (2, -\frac{5}{2}) \), making it a single point in the Cartesian plane. This equation represents a specific case where we're looking at a circle that has degenerated into a single point. In real-world applications, this might represent a position where two variables must meet certain criteria, such as in a coordinate system for an engineering project or in optimization scenarios where only one optimal solution exists.
