document.write( "Question 748171: find the quadratic function which takes the value 41 at x=-2 and the value 20 at x=5 and is minimized at x=21
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Algebra.Com's Answer #455369 by Edwin McCravy(20055)\"\" \"About 
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find the quadratic function which takes the value 41 at x=-2
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document.write( " y = Ax²+Bx+C\r\n" );
document.write( "41 = A(-2)²+B(-2)+C\r\n" );
document.write( "41 = A(4)-2B+C\r\n" );
document.write( "41 = 4A-2B+C\r\n" );
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document.write( "first equation:   4A-2B+C = 41\r\n" );
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document.write( " y = Ax²+Bx+C\r\n" );
document.write( "20 = A(5)²+B(5)+C\r\n" );
document.write( "20 = A(25)+5B+C\r\n" );
document.write( "20 = 25A+5B+C\r\n" );
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document.write( "second equation:   25A-5B+C = 20\r\n" );
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document.write( "A parabola has its lowest point, that is, it is\r\n" );
document.write( "minimized at its vertex.  Its vertex has as its \r\n" );
document.write( "x-coordinate the value \"-B%2F%282A%29\" so we set \r\n" );
document.write( "that = 21\r\n" );
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document.write( "\"-B%2F%282A%29\" = 21\r\n" );
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document.write( "Cross-multiplying:\r\n" );
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document.write( "      -B = 42A\r\n" );
document.write( "-42A - B = 0\r\n" );
document.write( " 42A + B = 0\r\n" );
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document.write( "third equation:    42A + B = 0\r\n" );
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document.write( "eq. 1    4A-2B+C = 41\r\n" );
document.write( "eq. 2   25A+5B+C = 20\r\n" );
document.write( "eq. 3   42A+ B   = 0\r\n" );
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document.write( "Multiply eq. 1 by -1 and add it to eq. 2 to eliminate\r\n" );
document.write( "the C's\r\n" );
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document.write( "        25A+5B+C =  20\r\n" );
document.write( "        -4A+2B-C = -41\r\n" );
document.write( "      -----------------\r\n" );
document.write( "        21A+7B   = -21\r\n" );
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document.write( "That can be easily divided through by 7\r\n" );
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document.write( "eq. 4    3A+ B   = -3\r\n" );
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document.write( "Now we have a system with only 2 equations and\r\n" );
document.write( " 2 unknowns.\r\n" );
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document.write( "eq. 4    3A+ B = -3\r\n" );
document.write( "eq. 3   42A+ B =  0\r\n" );
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document.write( "Multiply eq. 4 through by -1 and add to eq. 3 to ekiminate\r\n" );
document.write( "the B's.\r\n" );
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document.write( "        -3A- B = 3\r\n" );
document.write( "        42A+ B = 0\r\n" );
document.write( "        ----------\r\n" );
document.write( "        39A    = 3\r\n" );
document.write( "             A = \"3%2F39\"\r\n" );
document.write( "             A = \"1%2F13\" \r\n" );
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document.write( "Since that's a nasty fraction, it will be easier to start over\r\n" );
document.write( "and eliminate A from eqs. 4 and 3\r\n" );
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document.write( "3 goes into 42 14 times so we just have to multiply eq. 4 by\r\n" );
document.write( "-14 and add it to eq. 3:\r\n" );
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document.write( "       -42A-14B = 42\r\n" );
document.write( "        42A+  B =  0\r\n" );
document.write( "       --------------\r\n" );
document.write( "           -13B = 42\r\n" );
document.write( "              B = \"%2842%29%2F%28-13%29\"\r\n" );
document.write( "              B = \"-42%2F13\"\r\n" );
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document.write( "Substitute that and A = \"1%2F13\" into eq. 1: \r\n" );
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document.write( "eq. 1    4A-2B+C = 41\r\n" );
document.write( "      4\"%281%2F13%29\"-2\"%28-42%2F13%29\"+C = 41\r\n" );
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document.write( "Clear of fractions by multiplying through by 13\r\n" );
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document.write( "        4 + 84 + 13C = 533\r\n" );
document.write( "            88 + 13C = 533\r\n" );
document.write( "                 13C = 445\r\n" );
document.write( "                   C = \"445%2F13\"\r\n" );
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document.write( "Those are terrible fractions, but they are correct.  I\r\n" );
document.write( "suspect you may have made a slight typo, such as typing\r\n" );
document.write( "a sign or a digit wrong in one of your figures. \r\n" );
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document.write( "But the correct function based on the numbers you gave, is:\r\n" );
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document.write( "y = \"1%2F13\"x² - \"42%2F13\"x + \"445%2F13\"\r\n" );
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document.write( "or if you like you can use functional notation f(x) for y:\r\n" );
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document.write( "f(x) = \"1%2F13\"x² - \"42%2F13\"x + \"445%2F13\"\r\n" );
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document.write( "Edwin
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