Question 1179452
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x<sup>2</sup> + y<sup>2</sup> + 6x + 4y = 3

Get the x-terms next to each other.
Get the y-terms next to each other.
Skip a space after the x terms and after the y terms:

x<sup>2</sup> + 6x     + y<sup>2</sup> + 4y     = 3 

Complete the square out to the side or on scratch paper.

1. Multiply the coefficient of x by 1/2.   {{{6*expr(1/2)=3}}}
2. Square what you get.                    {{{3^2=9}}}
3. Add that to the space after the x-terms on the left and also
   add it to the right side.

x<sup>2</sup> + 6x + 9 + y<sup>2</sup> + 4y     = 3 + 9 

1. Multiply the coefficient of y by 1/2.   {{{4*expr(1/2)=2}}}
2. Square what you get.                    {{{2^2=4}}}
3. Add that to the space after the y-terms on the left and also
   add it to the right side.

x<sup>2</sup> + 6x + 9 + y<sup>2</sup> + 4y + 4 = 3 + 9 + 4

Factor the first three terms on the left side.
Factor the last three terms on the left side.
Combine the numbers on the right side.

(x + 3)(x + 3) + (y + 2)(y + 2) = 16

Notice that they factored as the same factor twice, so write
them as squares:

(x + 3)<sup>2</sup> + (y + 2)<sup>2</sup> = 16

Compare that to what you should have memorized.

(x - h)<sup>2</sup> + (y - k)<sup>2</sup> = r<sup>2</sup> where (h,k) is the center and r is the radius.

Set corresponding things equal to each other and solve

-h = +3        -k = +2       r<sup>2</sup> = 16
 h = -3         k = -2        r = √16
                              r = 4

Substitute for all letters except x and y

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

Simplify:

{{{(x+3)^2+(y+2)^2=16}}}

Center = (h,k) = (-3,-2),  radius = r = 4

{{{drawing(400,400,-10,10,-10,10, grid(1),circle(-3,-2,4), circle(-3,-2,.1) )}}}

Edwin</pre>