x² + 2y² + 6x - 20y + 53 = 0
Rearrange the terms as
x² + 6x + 2y² - 20y = -53
Complete the square of the first two terms:
Multiply 6 by getting 3
Square 3, getting +9
Add +9 to both sides
x² + 6x + 9 + 2y² - 20y = -53+9
Factor the first three terms x²+6x+9 as (x+3)(x+3) or (x+3)²
and combine the terms on the right:
(x + 3)² + 2y² - 20y = -44
Factor 2 out of the last two terms on the left:
(x + 3)² + 2y² - 20y = -44
(x + 3)² + 2(y² - 10y) = -44
Complete the square of the two terms inside the
second parentheses:
Multiply -10 by getting -5
Square -5, getting +25
Add +25 inside the second parentheses and since there
is a 2 coefficient, we are really adding 2 times +25 or
+50 to both sides, so we add +50 to the right sides:
(x + 3)² + 2(y² - 10y + 25) = -44 + 50
Factor the three terms y²-10y+25 as (y-5)(y-5) or (y-5)²
and combine the terms on the right:
(x + 3)² + 2(y - 5)² = 6
Get 1 on the right by dividing all terms by 6
+ = + = 1
Edwin
