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 bygetting 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