Question 625240
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16x² + 4y² + 96x - 8y + 84 = 0

Get the constant term off the left side by adding -84 to both sides

   16x² + 96x + 4y² - 8y = -84

Swap the two middle terms

   16x² + 96x + 4y² - 8y = -84

Factor out the coefficients of x² and y² 

16(x² + 6x) + 4(y² - 2y) = -84

To complete the square inside the first parenhtheses,
1. Multiply the coefficient of x, which is +6 by 1/2, getting 3
2. Square this result, (3)² = +9
3. Add +9 inside the first parentheses
4. Multiply +9 by the coefficient we factored out, 16, getting +144
5. Add +144 to the right side  

16(x² + 6x + 9) + 4(y² - 2y) = -84 + 144

To complete the square inside the second parenhtheses,
1. Multiply the coefficient of y, which is -2 by 1/2, getting -1
2. Square this result, (-1)² = +1
3. Add +9 inside the first parentheses
4. Multiply +1 by the coefficient we factored out, 4, getting +4
5. Add +4 to the right side  

16(x² + 6x + 9) + 4(y² - 2y + 1) = -84 + 144 + 4

Factor the 1st parentheses:  x² + 6x + 9 = (x + 3)(x + 3) = (x + 3)²
Factor the 2nd parentheses:  y² - 2x + 1 = (y - 1)(y - 1) = (y - 1)²
Combine terms on the right:  -84 + 144 + 4 = 64

16(x + 3)² + 4(y - 1)² = 64

Get a 1 on the right side by dividing through by 64

{{{16(x + 3)^2/64}}} + {{{4(y - 1)^2/64}}} = {{{64/64}}}

{{{(x + 3)^2/4}}} + {{{(y - 1)^2/16}}} = 1

{{{(x + 3)^2/2^2}}} + {{{(y - 1)^2/4^2}}} = 1

To translate this equation 6 units down and 7 units to the left,
replace y by (y+6) and x by (x+7)

{{{((x+7) + 3)^2/2^2}}} + {{{((y+6) - 1)^2/4^2}}} = 1

{{{(x+7 + 3)^2/2^2}}} + {{{(y+6 - 1)^2/4^2}}} = 1

{{{(x+10)^2/2^2}}} + {{{(y+5)^2/4^2}}} = 1

Edwin</pre>