document.write( "Question 122301: Find all real or imaginary solution to the question.\r
\n" ); document.write( "\n" ); document.write( "6x=-(19x+25)/x+1 \n" ); document.write( "
Algebra.Com's Answer #89795 by solver91311(24713)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "I'm going to assume you meant\r
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document.write( "\"6x=-%2819x%2B25%29%2F%28x%2B1%29\"\r
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document.write( "rather than\r
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document.write( "\"6x=-%2819x%2B25%29%2Fx%2B1\"\r
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document.write( "First make a note to exclude \"x=-1\" if it happens to come up as a potential root of the equation because -1 is not in the domain of the right hand side of the equation because -1 would make the denominator equal zero.\r
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document.write( "Second, move everything to the left of the equal sign, thus:\r
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document.write( "\"6x%2B%2819x%2B25%29%2F%28x%2B1%29=0\"\r
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document.write( "\"x%2B1\" is your common denominator, so:\r
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document.write( "\"%286x%28x%2B1%29%2B%2819x%2B25%29%29%2F%28x%2B1%29=0\"\r
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document.write( "Simplify the numerator
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document.write( "\"%286x%5E2%2B6x%2B19x%2B25%29%2F%28x%2B1%29=0\"
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document.write( "\"%286x%5E2%2B25x%2B25%29%2F%28x%2B1%29=0\"\r
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document.write( "Remember that \"a%2Fb=0\" if and only if \"a=0\" and \"b%3C%3E0\", meaning that we need to solve for:\r
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document.write( "\"6x%5E2%2B25x%2B25=0\"\r
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document.write( "Oddly enough, this mess actually factors.\r
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document.write( "\"%283x%2B5%29%2A%282x%2B5%29=0\", hence\r
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document.write( "\"3x%2B5=0\"
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document.write( "\"x=-5%2F3\"\r
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document.write( "or\r
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document.write( "\"2x%2B5=0\"
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document.write( "\"x=-5%2F2\"\r
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document.write( "Since neither of these roots = -1, both are valid potential roots of the
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document.write( "equation.  However, since we introduced an \"x%5E2\" term by applying the
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document.write( "common denominator, we need to check both roots against the possiblity that
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document.write( "one of them is extraneous.\r
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document.write( "\"6%28-5%2F3%29=-%2819%28-5%2F3%29%2B25%29%2F%28-5%2F3%2B1%29\"
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document.write( "\"-10=-%28-%2895%2F3%29%2B75%2F3%29%2F%28-2%2F3%29\"
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document.write( "\"-10=-%28-20%2F3%29%2F%28-2%2F3%29\"
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document.write( "\"-10=-%28-20%29%2F%28-2%29\"
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document.write( "\"-10=-20%2F2\"  First root checks.\r
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document.write( "\"6%28-5%2F2%29=-%2819%28-5%2F2%29%2B25%29%2F%28-5%2F2%2B1%29\"
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document.write( "\"-15=-%28-%2895%2F2%29%2B50%2F2%29%2F%28-3%2F2%29\"
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document.write( "\"-15=-%28-45%2F2%29%2F%28-3%2F2%29\"
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document.write( "\"-15=-%28-45%29%2F%28-3%29\"
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document.write( "\"-15=-45%2F3\"  Second root checks.\r
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document.write( "Done
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