Question
Upstudy AI Solution
Tutor-Verified Answer
Answer
The equation simplifies to \( (x + 1)^2 + (y + 1)^2 = 61 \), which is a circle centered at (-1, -1) with a radius of √61.
Solution
We start with the equation:
x² + 2x + y² + 2y = 59
Step 1. Group the x-terms and y-terms:
(x² + 2x) + (y² + 2y) = 59
Step 2. Complete the square for each group.
For x² + 2x:
Take half of 2, which is 1, and square it to get 1.
Add and subtract 1:
(x² + 2x + 1) - 1 = (x + 1)² - 1
For y² + 2y:
Take half of 2, which is 1, and square it to get 1.
Add and subtract 1:
(y² + 2y + 1) - 1 = (y + 1)² - 1
Step 3. Substitute these back into the original equation:
(x + 1)² - 1 + (y + 1)² - 1 = 59
Step 4. Simplify the equation:
(x + 1)² + (y + 1)² - 2 = 59
(x + 1)² + (y + 1)² = 61
Final Answer:
The equation in standard form is:
(x + 1)² + (y + 1)² = 61
This represents a circle with center (-1, -1) and radius √61.
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution Mind Expander
This equation represents a circle in the \(xy\)-plane. If we rearrange it, we can complete the square for both the \(x\) and \(y\) terms. This leads us to \( (x+1)^{2} + (y+1)^{2} = 59 - 2 - 2 = 55 \). With its center at \((-1, -1)\) and a radius of \(\sqrt{55}\), it gives us a neat geometric visualization! Moreover, you can see how such equations are useful in various fields. For instance, in physics, circular motion is often described using equations of circles. Knowing how to analyze such equations can help one solve complex problems in kinematics and even in video game design where movement can be modeled using circular paths.
