To find the equation of the given curve, we need to rewrite it in a standard form. Let's simplify the equation step by step:
1. (X-1)^2 + (y+3) = 4
2. (X-1)^2 = 4 - (y+3)
3. (X-1)^2 = 1 - (y+3) [Since 4 - (y+3) = 1]
4. (X-1)^2 = 1 - y - 3
5. (X-1)^2 = -2 - y
Now, to make it more familiar, we can write it in terms of y:
6. y = -2 - (X-1)^2
So, the equation of the curve is y = -2 - (X-1)^2. This equation represents a parabola that opens downward with its vertex at (1, -2).