To find the x-coordinate of the center of the circular mosaic given by the equation \( x^2 + y^2 + 4x - 6y = 0 \), we first need to rewrite this equation in standard form by completing the square.
-
Start with the given equation: \[ x^2 + y^2 + 4x - 6y = 0 \]
-
Rearrange the equation: \[ x^2 + 4x + y^2 - 6y = 0 \]
-
Complete the square for \( x \):
- For \( x^2 + 4x \), take half of 4, square it, giving \( 4 \): \[ x^2 + 4x = (x + 2)^2 - 4 \]
-
Complete the square for \( y \):
- For \( y^2 - 6y \), take half of -6, square it, giving \( 9 \): \[ y^2 - 6y = (y - 3)^2 - 9 \]
-
Substitute these back into the equation: \[ (x + 2)^2 - 4 + (y - 3)^2 - 9 = 0 \] \[ (x + 2)^2 + (y - 3)^2 - 13 = 0 \] \[ (x + 2)^2 + (y - 3)^2 = 13 \]
Now we have the equation in standard form, which describes a circle: \[ (x + 2)^2 + (y - 3)^2 = 13 \] From this equation, we can identify the center of the circle at the coordinates \( (-2, 3) \).
Therefore, the x-coordinate of the center is: \[ \text{x-coordinate} = -2 \]
So the answer is \(-2\).