document.write( "Question 386132: Graph the equation \r
\n" ); document.write( "\n" ); document.write( "2x^2 + 2y^2 -16x + 4y - 38 = 0 \n" ); document.write( "

Algebra.Com's Answer #273018 by jsmallt9(3758) $\"\"$ $\"About$

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$\"2x%5E2+%2B+2y%5E2+-16x+%2B+4y+-+38+=+0\"$
\n" ); document.write( "Since everything is divisible by 2, I am going to start by dividing both sides by 2. This will create smaller numbers and coefficients of 1 in front of the squared terms. This will make the rest of the problem easier:
\n" ); document.write( " $\"x%5E2+%2B+y%5E2+-8x+%2B+2y+-+19+=+0\"$
\n" ); document.write( "With no xy term and with equal coefficients in front of the squared terms, this equation is the equation of a circle. So we want it in the general form for the equation of a circle:
\n" ); document.write( " $\"%28x-h%29%5E2+%2B+%28y-k%29%5E2+=+r%5E2\"$
\n" ); document.write( "So we need to complete the squares for both the x terms and the y terms. When completing squares I like to start by \"moving\" the constant term to the other side of the equation. So I'll add 19 to each side (and rearrange the terms so the x terms and y terms are together):
\n" ); document.write( " $\"x%5E2+-8x+%2B+y%5E2++%2B+2y+=+19\"$
\n" ); document.write( "The next step is to figure out what constant terms are needed to make

$\"x%5E2+-8x\"$ a perfect square trinomial, and
$\"y%5E2+%2B+2y\"$ a perfect square trinomial.

\n" ); document.write( "To find these constant terms we take half of the coeeficient of x (or y) and square it. Half of -8 is -4. -4 squared is 16. So we need to add 16 to each side to complete the square for the x terms. For the y terms, half of 2 is 1 and 1 squared is 1 so we need to add 1 to each side to complete the square for the y terms:
\n" ); document.write( " $\"x%5E2+-8x+%2B+16+%2B+y%5E2++%2B+2y+%2B+1=+19+%2B+16+%2B+1\"$
\n" ); document.write( "On the left side we can rewrite the equation as two perfect squares and on the right side we just add up the numebers:
\n" ); document.write( " $\"%28x-4%29%5E2+%2B+%28y%2B1%29%5E2+=+36\"$
\n" ); document.write( "The only things left to do are to write the perfect square for the y's as a subtraction and to write the right side as a perfect square:
\n" ); document.write( " $\"%28x-4%29%5E2+%2B+%28y-+%28-1%29%29%5E2+=+6%5E2\"$
\n" ); document.write( "Now the equation is in the desired form for a circle. We can read the h and the k which are coordinates of the center of the circle and the r which is the radius:
\n" ); document.write( "h = 4
\n" ); document.write( "k = -1
\n" ); document.write( "Center: (4, -1)
\n" ); document.write( "r = radius = 6
\n" ); document.write( "With the center and the radius you should be able to sketch a graph of this circle. \n" ); document.write( "

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