document.write( "Question 189584: Can you please help me solve this? I'm supposed to solve for all roots but I'm not sure how. \"8x%5E3%2B27=0\" \n" ); document.write( "
Algebra.Com's Answer #142261 by Edwin McCravy(20059)\"\" \"About 
You can put this solution on YOUR website!
Can you please help me solve this? I'm supposed to solve for all roots but I'm not sure how. \"8x%5E3%2B27=0\"
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document.write( "You must learn the principle of factoring the sum and difference of\r\n" );
document.write( "two cubes:\r\n" );
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document.write( "\"FIRST%5E3+%2B+SECOND%5E3\" \r\n" );
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document.write( "factors as\r\n" );
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document.write( "\"%28FIRST%2BSECOND%29%28FIRST%5E2-FIRST%2ASECOND%2BSECOND%5E2%29\"\r\n" );
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document.write( "and\r\n" );
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document.write( "\"FIRST%5E3+-+SECOND%5E3\" \r\n" );
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document.write( "factors as\r\n" );
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document.write( "\"%28FIRST-SECOND%29%28FIRST%5E2%2BFIRST%2ASECOND%2BSECOND%5E2%29\"\r\n" );
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document.write( "Your problem is the first case as it has a + sign between\r\n" );
document.write( "the two terms:\r\n" );
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document.write( "\"8x%5E3%2B27=0\"\r\n" );
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document.write( "Notice that \"8x%5E3\" can be written as \"%282x%29%5E3\"\r\n" );
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document.write( "Notice that \"27\" can be written as \"%283%29%5E3\"\r\n" );
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document.write( "So we write it as \r\n" );
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document.write( "\"%282x%29%5E3%2B%283%29%5E3=0\"\r\n" );
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document.write( "So \"FIRST=2x\" and \"SECOND=3\"\r\n" );
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document.write( "and since\r\n" );
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document.write( "\"FIRST%5E3+%2B+SECOND%5E3\" \r\n" );
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document.write( "factors as\r\n" );
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document.write( "\"%28FIRST%2BSECOND%29%28FIRST%5E2-FIRST%2ASECOND%2BSECOND%5E2%29\"\r\n" );
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document.write( "we factor the left side of \r\n" );
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document.write( "\"%282x%29%5E3%2B%283%29%5E3=0\"\r\n" );
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document.write( "as\r\n" );
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document.write( "\"%282x%2B3%29%28%282x%29%5E2-%282x%29%2A%283%29%2B%283%29%5E2%29=0%29\"\r\n" );
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document.write( "Simplifying inside the second parentheses:\r\n" );
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document.write( "\"%282x%2B3%29%28%282%5E2%29%28x%5E2%29-%286x%29%2B9%29=0\"\r\n" );
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document.write( "\"%282x%2B3%29%284x%5E2-6x%2B9%29=0\"\r\n" );
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document.write( "So we use the zero factor property:\r\n" );
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document.write( "\"2x%2B3=0\",  \"4x%5E2-6x%2B9=0\"\r\n" );
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document.write( "Solving the first,\r\n" );
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document.write( "\"2x%2B3=0\"\r\n" );
document.write( "\"2x=-3\"\r\n" );
document.write( "\"x=-3%2F2\"\r\n" );
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document.write( "So one root is \"-3%2F2\" \r\n" );
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document.write( "Solving the second:\r\n" );
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document.write( "\"4x%5E2-6x%2B9=0\"\r\n" );
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document.write( "That doesn't factor so we resort to the\r\n" );
document.write( "quadratic formula:\r\n" );
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document.write( "\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\" \r\n" );
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document.write( "where \"a=4\", \"b=-6\", \"c=9\"\r\n" );
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document.write( "\"x+=+%28-%28-6%29+%2B-+sqrt%28+%28-6%29%5E2-4%2A%284%29%2A%289%29+%29%29%2F%282%2A%284%29%29+\" \r\n" );
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document.write( "\"x+=+%286+%2B-+sqrt%2836-144%29+%29%2F8+\" \r\n" );
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document.write( "\"x+=+%286+%2B-+sqrt%28-108%29+%29%2F8+\"\r\n" );
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document.write( "\"x+=+%286+%2B-+i%2Asqrt%28108%29+%29%2F8+\"\r\n" );
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document.write( "\"x+=+%286+%2B-+i%2Asqrt%2836%2A3%29+%29%2F8+\"\r\n" );
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document.write( "\"x+=+%286+%2B-+6i%2Asqrt%283%29+%29%2F8+\"\r\n" );
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document.write( "Make two fractions on the right\r\n" );
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document.write( "\"x+=+6%2F8+%2B-+%286i%2Asqrt%283%29%29%2F8\"\r\n" );
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document.write( "Reduce the fractions by dividing top and \r\n" );
document.write( "bottom by 2:\r\n" );
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document.write( "\"x+=+3%2F4+%2B-+%283i%2Asqrt%283%29%29%2F4\"\r\n" );
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document.write( "So the other two roots are\r\n" );
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document.write( "\"3%2F4+%2B+%283sqrt%283%29%29%2F4\"\"i\"\r\n" );
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document.write( "and \r\n" );
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document.write( "\"3%2F4+-+%283sqrt%283%29%29%2F4\"\"i\"\r\n" );
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document.write( "Edwin

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