document.write( "Question 1209082: Real numbers a and b satisfy
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Algebra.Com's Answer #847681 by mccravyedwin(407)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "\"system%28a+%2B+ab%5E2+=+250b%2C%0D%0Aa+%2B+b+=+102%29\"\r\n" );
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document.write( "Solve the second for b = (102-a) and substitute in the first:\r\n" );
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document.write( "\"a+%2B+a%28102-a%29%5E2+=+250%28102-a%29\"\r\n" );
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document.write( "\"a+%2B+a%2810404-204a%2Ba%5E2%29=25500-250a\"\r\n" );
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document.write( "\"a+%2B+10404a-204a%5E2%2Ba%5E3=25500-250a\"\r\n" );
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document.write( "\"a%5E3-204a%5E2%2B10655a-25500=0\"\r\n" );
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document.write( "Since the last term is divisible by 100, let's take a chance and\r\n" );
document.write( "see if 100 is a root with synthetic division:\r\n" );
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document.write( "100 | 1 -204  10655 -25500\r\n" );
document.write( "    |    100 -10400  25500\r\n" );
document.write( "      1 -104    255      0\r\n" );
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document.write( "Sure enough, 100 is a root! So the cubic factors as\r\n" );
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document.write( "\"%28a-100%29%28a%5E2-104a%2B255%29=0\"\r\n" );
document.write( "a-100=0;  a2-104a+255=0\r\n" );
document.write( "    a=100; \r\n" );
document.write( "  b=102-a;\r\n" );
document.write( "b=102-100=2\r\n" );
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document.write( "So there is one solution: a=100, b=2. (a,b) = (100,2)\r\n" );
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document.write( "Checking:\r\n" );
document.write( "\"system%28a+%2B+ab%5E2+=+250b%2C%0D%0Aa+%2B+b+=+102%29\"\r\n" );
document.write( "\"system%28100+%2B+%28100%29%282%29%5E2+=+250%282%29%2C%0D%0A100+%2B+2+=+102%29\"\r\n" );
document.write( "\"system%28100+%2B+400+=+500%2C%0D%0A102=+102%29\"\r\n" );
document.write( "That checks.  But that cubic equation has too more solutions.\r\n" );
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document.write( "You know how to solve a quadratic by the quadratic formula,\r\n" );
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document.write( "So solve this quadratic equation by the quadratic formula as\r\n" );
document.write( "it doesn't factor:\r\n" );
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document.write( "\"a%5E2-104a%2B255=0\"\r\n" );
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document.write( "and you'll get two more solutions for \" a \",\r\n" );
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document.write( "Then substitute them in b = 102-a and you'll have two more \r\n" );
document.write( "solutions (a,b).\r\n" );
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document.write( "Edwin
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