Question

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

Answer 1

2*(x - 3)² + 2*(y + 5)² = 96.

Explanation:

Answer 2

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

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