Question

Complete the square x^2+y^2+4x-6y+1=0

Answer 1

To complete the square for the given equation x^2 + y^2 + 4x - 6y + 1 = 0, we need to rearrange the terms and group the x and y terms separately:

(x^2 + 4x) + (y^2 - 6y) + 1 = 0

Now, we need to find the values that complete the squares for x and y.

For x^2 + 4x, we add (4/2)^2 = 4 to complete the square:
(x + 2)^2 = x^2 + 4x + 4

For y^2 - 6y, we add (-6/2)^2 = 9 to complete the square:
(y - 3)^2 = y^2 - 6y + 9

Now, we can rewrite the equation in terms of completed squares:
(x + 2)^2 - 4 + (y - 3)^2 - 9 + 1 = 0
(x + 2)^2 + (y - 3)^2 - 12 = 0

Therefore, the equation x^2 + y^2 + 4x - 6y + 1 = 0 can be rewritten as (x + 2)^2 + (y - 3)^2 - 12 = 0.