document.write( "Question 474815: I need a brush-up on completing the square. I am working with equations for ellipses. I just need help getting this problem into the right form for an ellipse by completing the square.\r
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\n" ); document.write( "\n" ); document.write( "x^2-2x+4y^2+8y+1=0\r
\n" ); document.write( "\n" ); document.write( "I know that eventually this ends up looking like (x^2-2x+1)+4(y^2+2y+1)=-1+1+4(1). I do not understand how we get here. I know that we are supposes to multiply both(?) sides by the square of half of the x coefficients, but I'm still confused. Anyway, an explanation for a dummy like me would be super awesome. Thanks! \n" ); document.write( "
Algebra.Com's Answer #325621 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "x²-2x+4y²+8y+1 = 0\r\n" );
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document.write( "The object is to make it look like this:\r\n" );
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document.write( " (x-h)²  (y-k)²    \r\n" );
document.write( "—————— + ——————— = 1\r\n" );
document.write( "   a²      b²\r\n" );
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document.write( "which is an ellipse that looks like this \"ᆼ\"\r\n" );
document.write( "like an egg on a table, or:\r\n" );
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document.write( " (x-h)²  (y-k)²    \r\n" );
document.write( "—————— + ——————— = 1\r\n" );
document.write( "   b²      a²\r\n" );
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document.write( "which is an ellipse that looks like the number \"0\". \r\n" );
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document.write( "We can tell because a² > b²\r\n" );
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document.write( "We start with this:\r\n" );
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document.write( "x²-2x+4y²+8y+1 = 0\r\n" );
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document.write( "We get the 1 on the other side as a -1:\r\n" );
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document.write( "x²-2x+4y²+8y = -1\r\n" );
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document.write( "It's good that the terms are in the order they are.\r\n" );
document.write( "Sometimes we have to switch them so that the terms\r\n" );
document.write( "are in the order \"x², x, y², y\" as they are here.\r\n" );
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document.write( "Write it this way: \r\n" );
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document.write( "[x²-2x]+[4y²+8y] = -1 \r\n" );
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document.write( "The coefficient of x² is 1 so we don't do anything yet to\r\n" );
document.write( "the first bracket. \r\n" );
document.write( "But the coefficient of y² is 4, so let's factor that out\r\n" );
document.write( "in the 2nd bracket.\r\n" );
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document.write( "[x²-2x]+[4(y²+2y)] = -1\r\n" );
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document.write( "Now I'll dispense with the brackets and just have parentheses:\r\n" );
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document.write( "(x²-2x)+4(y²+2y) = -1\r\n" );
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document.write( "Next we want to make those two binomials into trinomials.\r\n" );
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document.write( "We skip some space after those binomials \r\n" );
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document.write( "(x²-2x  )+4(y²+2y  ) = -1\r\n" );
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document.write( "so we can add a number in those two boxes to make those \r\n" );
document.write( "binomials into trinomials so they'll factor into squares \r\n" );
document.write( "of binomials.\r\n" );
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document.write( "(x²-2x+↏)+4(y²+2y+↏) = -1\r\n" );
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document.write( "Now let's figure out what number goes in the first box.\r\n" );
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document.write( "The coefficient of x is -2 so we take half of it, getting -1,\r\n" );
document.write( "then we square -1, getting (-1)² or 1,\r\n" );
document.write( "so we add 1 where the first box is, but we also have to add\r\n" );
document.write( "1 to the other side of the equation, like this:\r\n" );
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document.write( "[x²-2x+4]+4(y²+2y+↏) = -1+1\r\n" );
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document.write( "Now let's figure out what number goes in the second box.\r\n" );
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document.write( "The coefficient of y is 2 so we take half of it, getting 1,\r\n" );
document.write( "then we square 1, getting 1² or 1, but wait!  See the 4 in \r\n" );
document.write( "front of the second parentheses?  If we put a 1 in that box,\r\n" );
document.write( "It will get multiplied by the 4 in front of the parentheses.\r\n" );
document.write( "In other words putting a 1 in that second box will in effect \r\n" );
document.write( "amount to the same as adding 4 times 1 or 4 to the left side,\r\n" );
document.write( "not just 1.  So we have to add 4(1) to the right side to offset\r\n" );
document.write( "adding 1 inside that parentheses since it will be multiplied\r\n" );
document.write( "by the 4, so we have:\r\n" );
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document.write( "(x²-2x+4)+4(y²+2y+1) = -1+1+4(1)\r\n" );
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document.write( "Notice that what's in the first parentheses,\r\n" );
document.write( "x²-2x+4 factors as (x-2)(x-2) or (x-2)²\r\n" );
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document.write( "Also notice that what's in the second parentheses\r\n" );
document.write( "y²+2y+1 factors as (y+1)(y+1) or (y+1)².\r\n" );
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document.write( "So this\r\n" );
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document.write( "(x²-2x+4)+4(y²+2y+1) = -1+1+4(1)\r\n" );
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document.write( "after substituting their factorization for the parentheses\r\n" );
document.write( "and combining the terms on the right, we have\r\n" );
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document.write( "(x-2)²+4(y+1)² = 4\r\n" );
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document.write( "Next we get a 1 on the right by dividing all three terms by 4:\r\n" );
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document.write( "(x-2)²   4(y+1)²    4\r\n" );
document.write( "—————— + ——————— = ——— \r\n" );
document.write( "  4         4       4\r\n" );
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document.write( "And that simplifies to:\r\n" );
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document.write( " (x-2)²  (y+1)²    \r\n" );
document.write( "—————— + ——————— = 1\r\n" );
document.write( "   4       1       \r\n" );
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document.write( "which is in the form:\r\n" );
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document.write( " (x-h)²  (y-k)²    \r\n" );
document.write( "—————— + ——————— = 1\r\n" );
document.write( "   a²      b²\r\n" );
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document.write( "because a² = 4 and b² = 1\r\n" );
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document.write( "The center is (h,k) = (2,-1)\r\n" );
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document.write( "Plot it:\r\n" );
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document.write( "Since a² = 4, a = 2\r\n" );
document.write( "Since b² = 1, b = 1\r\n" );
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document.write( "a = 2 is the semi-major axis's length, so draw the\r\n" );
document.write( "major axis 2a or 4 units long with the center as the midpoint.\r\n" );
document.write( "We also draw the minor axis 2a or 4 units long with the center \r\n" );
document.write( "as the midpoint:\r\n" );
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document.write( "Now we can sketch in the ellipse:\r\n" );
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document.write( "The vertices are the endpoints of the major axes,\r\n" );
document.write( "(0,-1) and (4,-1).\r\n" );
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document.write( "The covertices are the endpoints of the minor\r\n" );
document.write( "axis, (2,0) and (2,-2).\r\n" );
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document.write( "Edwin
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