document.write( "Question 418019: Factor:72+22x-x² \n" ); document.write( "
Algebra.Com's Answer #292605 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "The other tutor is right in that it does not factor.  But\r\n" );
document.write( "the quadratic formula can only be used to solve an equation.\r\n" );
document.write( "This is not an equation, but an expression, specifically a\r\n" );
document.write( "quadratic trinomial, not an equation.  Let's see why it\r\n" );
document.write( "doesn't factor.  Let's try anyway:\r\n" );
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document.write( "72+22x-x²\r\n" );
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document.write( "We write it in descending order:\r\n" );
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document.write( "-x²+22x+72\r\n" );
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document.write( "Factor out -1\r\n" );
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document.write( "-1(x²-22x-72)\r\n" );
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document.write( "We look for factor pairs of the last term in\r\n" );
document.write( "absolute value, 72\r\n" );
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document.write( "72*1=72\r\n" );
document.write( "36*2=72\r\n" );
document.write( "24*3=72\r\n" );
document.write( "18*4=72\r\n" );
document.write( "12*6=72\r\n" );
document.write( " 9*8=72\r\n" );
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document.write( "Since the sign of the last term is -,\r\n" );
document.write( "out beside them we SUBTRACT them. \r\n" );
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document.write( "72*1=72   72-1=71\r\n" );
document.write( "36*2=72   36-2=34\r\n" );
document.write( "24*3=72   24-3=21\r\n" );
document.write( "18*4=72   18-4=14\r\n" );
document.write( "12*6=72   12-6= 6\r\n" );
document.write( " 9*8=72    9-8= 1 \r\n" );
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document.write( "We do not find the middle term's coefficient in\r\n" );
document.write( "absolute value, 22, among those differences. \r\n" );
document.write( "Therefore the trinomial does not factor.\r\n" );
document.write( "It is what is called \"a prime polynomial\".\r\n" );
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document.write( "If the trinomial had been this instead\r\n" );
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document.write( "-72+22x-x²\r\n" );
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document.write( "it would have factored.\r\n" );
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document.write( "We write it in descending order:\r\n" );
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document.write( "-x²+22x-72\r\n" );
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document.write( "Factor out -1\r\n" );
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document.write( "-1(x²-22x+72)\r\n" );
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document.write( "We look for factor pairs of the last term in\r\n" );
document.write( "absolute value, 72\r\n" );
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document.write( "72*1=72\r\n" );
document.write( "36*2=72\r\n" );
document.write( "24*3=72\r\n" );
document.write( "18*4=72\r\n" );
document.write( "12*6=72\r\n" );
document.write( " 9*8=72\r\n" );
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document.write( "Since the sign of the last term is +,\r\n" );
document.write( "out beside them we ADD them. \r\n" );
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document.write( "72*1=72   72+1=74\r\n" );
document.write( "36*2=72   36+2=38\r\n" );
document.write( "24*3=72   24+3=27\r\n" );
document.write( "18*4=72   18+4=22\r\n" );
document.write( "12*6=72   12+6=18\r\n" );
document.write( " 9*8=72    9-8= 1 \r\n" );
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document.write( "For this problem, We do find the middle term's \r\n" );
document.write( "coefficient in absolute value, 22, among those differences,\r\n" );
document.write( "as indicated in red.  So we know that we can use the factor\r\n" );
document.write( "pair 18 and 4.\r\n" );
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document.write( "Therefore we write:\r\n" );
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document.write( "-1(x   18)(x   4)\r\n" );
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document.write( "Next we must fill in signs in the middle of each\r\n" );
document.write( "parentheses.  Since the sign of the middle term\r\n" );
document.write( "-22 is -, and since 18 is larger than 4, we place\r\n" );
document.write( "this sign before the larger, which is 18.\r\n" );
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document.write( "-1(x - 18)(x   4)\r\n" );
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document.write( "Since the sign of the last term is -, we place\r\n" );
document.write( "the opposite sign + before the 4\r\n" );
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document.write( "-1(x - 18)(x + 4)\r\n" );
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document.write( "So the expression you gave, 72+22x-x²\r\n" );
document.write( "does not factor, but -72+22x-x² does.\r\n" );
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document.write( "72+22x+x² would have factored as (x+18)(x+4)\r\n" );
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document.write( "72-22x+x² would have factored as (x-18)(x-4)\r\n" );
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document.write( "Could a sign have been inadvertently typed wrong?\r\n" );
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document.write( "Edwin
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