Which equation is equivalent to the following?

2x^2 - 12x + 2y^2 + 20y - 28 = 0

1) 2 (x-3) ^2 + 2 (y+5) ^2=44

2) 2 (x-3) ^2+2 (y+5) ^2=60

3) 2 (x-3) ^2+2 (y+5) ^2=62

4) 2 (x-3) ^2+2 (y+5) ^2=96

Therefore, the equivalent expression is: 2 * (x - 3) ² + 2 * (y + 5) ² = 96. Number 4)

Step-by-step explanation:

In order to find the equivalent expression, lets first factor the common term, which is "2":

2 * (x² - 6*x + y² + 10 * y - 14) = 0

If we look closely to the terms that have "x" and "y", they both have a square structure, therefore we can complete the squares by manipulating the indepent term. Let's first check the term with "x" and "y":

x² - 6*x + c² = x² - 2*3*x + c² = (x - c) ²

y² + 10*y + d² = y² + 2*5*y + d² = (y + d) ²

In order to complete the squares, c must be equal to 3 and d must be equal to 5. We can arrange that, by summing and subtracting these numbers squared, we have:

2 * (x² - 6*x + 3² + y² + 10 * y + 5² - 14 - 3² - 5²) = 0

2 * (x² - 6*x + 3²) + 2 * (y² + 10*y + 5²) - 28 - 18 - 50 = 0

2 * (x - 3) ² + 2 * (y + 5) ² = 96

Therefore, the equivalent expression is: 2 * (x - 3) ² + 2 * (y + 5) ² = 96