document.write( "Question 925613: If one root of the equation x^2-12x-P=0 is the square of the other, then find the value of P? \n" ); document.write( "
Algebra.Com's Answer #561620 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "That's one way of solving it.  Here's another.\r\n" );
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document.write( "Either way you'll have to check for extraneous solutions.\r\n" );
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document.write( "Let the two roots be x=a and x=b,\r\n" );
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document.write( "Then we have a system of three equations and 3 unknowns:\r\n" );
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document.write( "\"system%28a%5E2-12a-P=0%2Cb%5E2-12b-P=0%2Ca%5E2=b%29\"\r\n" );
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document.write( "Substitute \"a%5E2\" for \"b\" in\r\n" );
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document.write( "\"b%5E2-12b-P=0\"\r\n" );
document.write( "\"%28a%5E2%29%5E2-12%28a%5E2%29-P=0\"\r\n" );
document.write( "\"a%5E4-12a%5E2-P=0\"\r\n" );
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document.write( "Solve for P\r\n" );
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document.write( "\"a%5E4-12a%5E2=P\"\r\n" );
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document.write( "Substitute in\r\n" );
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document.write( "\"a%5E2-12a-P=0\"\r\n" );
document.write( "\"a%5E2-12a-%28a%5E4-12a%5E2%29=0\"\r\n" );
document.write( "\"a%5E2-12a-a%5E4%2B12a%5E2=0\"\r\n" );
document.write( "\"-a%5E4%2B13a%5E2-12a=0\"\r\n" );
document.write( "\"a%5E4-13a%5E2%2B12a=0\"\r\n" );
document.write( "\"a%28a%5E3-13a%2B12%29=0\"\r\n" );
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document.write( "It's easy to see that 1 is a solution to \"a%5E3-13a%2B12\"\r\n" );
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document.write( "1 | 1  0 -13  12\r\n" );
document.write( "  |    1   1 -12\r\n" );
document.write( "    1  1 -12   0\r\n" );
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document.write( "So we have factored the polynomial equation as\r\n" );
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document.write( "\"a%28a-1%29%28a%5E2%2Ba-12%29=0\"\r\n" );
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document.write( "We further factor it as\r\n" );
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document.write( "\"a%28a-1%29%28a%2B4%29%28a-3%29=0\"\r\n" );
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document.write( "So we have four solutions for a\r\n" );
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document.write( "a=0, a=1, a=-4, a=3\r\n" );
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document.write( "although one or more may be extraneous.\r\n" );
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document.write( "Substituting a=0 in\r\n" );
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document.write( "\"a%5E4-12a%5E2=P\"\r\n" );
document.write( "\"0%5E4-12%280%29%5E2=P\"\r\n" );
document.write( "\"0=P\"\r\n" );
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document.write( "Substituting a=1 in\r\n" );
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document.write( "\"a%5E4-12a%5E2=P\"\r\n" );
document.write( "\"1%5E4-12%281%29%5E2=P\"\r\n" );
document.write( "\"1-12=P\"\r\n" );
document.write( "\"-11=P\"\r\n" );
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document.write( "Substituting a=-4 in\r\n" );
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document.write( "\"a%5E4-12a%5E2=P\"\r\n" );
document.write( "\"%28-4%29%5E4-12%28-4%29%5E2=P\"\r\n" );
document.write( "\"256-12%2816%29=P\"\r\n" );
document.write( "\"256-192=P\"\r\n" );
document.write( "\"64=P\"\r\n" );
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document.write( "Substituting a=3 in\r\n" );
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document.write( "\"a%5E4-12a%5E2=P\"\r\n" );
document.write( "\"3%5E4-12%283%29%5E2=P\"\r\n" );
document.write( "\"81-12%289%29=P\"\r\n" );
document.write( "\"81-108=P\"\r\n" );
document.write( "\"-27=P\"\r\n" );
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document.write( "------------------\r\n" );
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document.write( "Checking:\r\n" );
document.write( "P=0\r\n" );
document.write( "\"x%5E2-12x-P=0\"\r\n" );
document.write( "\"x%5E2-12x-0=0\"\r\n" );
document.write( "\"x%5E2-12x=0\"\r\n" );
document.write( "\"x%28x-12%29=0\"\r\n" );
document.write( "\"x=0\", \"x=12\"\r\n" );
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document.write( "So P=0 is extraneous\r\n" );
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document.write( "P=-11\r\n" );
document.write( "\"x%5E2-12x-P=0\"\r\n" );
document.write( "\"x%5E2-12x-%28-11%29=0\"\r\n" );
document.write( "\"x%5E2-12x%2B11=0\"\r\n" );
document.write( "\"%28x-11%29%28x-1%29=0\"\r\n" );
document.write( "\"x=11\", \"x=1\"\r\n" );
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document.write( "So P=-11 is extraneous\r\n" );
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document.write( "P=64\r\n" );
document.write( "\"x%5E2-12x-P=0\"\r\n" );
document.write( "\"x%5E2-12x-64=0\"\r\n" );
document.write( "\"%28x%2B4%29%28x-16%29=0\"\r\n" );
document.write( "\"x=-4\", \"x=16\"\r\n" );
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document.write( "P=64 is a solution, since (-4)2=16\r\n" );
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document.write( "P=-27\r\n" );
document.write( "\"x%5E2-12x-P=0\"\r\n" );
document.write( "\"x%5E2-12x-%28-27%29=0\"\r\n" );
document.write( "\"x%5E2-12x%2B27=0\"\r\n" );
document.write( "\"%28x-3%29%28x-9%29=0\"\r\n" );
document.write( "\"x=3\", \"x=9\"\r\n" );
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document.write( "P=-27 is a solution, since 32=9\r\n" );
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document.write( "There are 2 actual solutions, P=64 and P=-27\r\n" );
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document.write( "Edwin
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