document.write( "Question 309868: find all solutions to x^3 - 6x +4 = 0. how many real and how many complex solutions? I can graph and get answers, but I'm not sure how to solve it otherwise. \n" ); document.write( "

Algebra.Com's Answer #221628 by Edwin McCravy(20055) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"x%5E3+-+6x+%2B4+=+0\"$
\n" ); document.write( "

\r\n" );
document.write( "All possible rational solutions are ą divisors of 4\r\n" );
document.write( "These are\r\n" );
document.write( "\r\n" );
document.write( "ą1, ą2, ą4\r\n" );
document.write( "\r\n" );
document.write( "We begin by trying 1. To use synthetic division we must \r\n" );
document.write( "consider the equation to be:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "1 | 1  0 -6  4\r\n" );
document.write( "  |    1  1 -5\r\n" );
document.write( "    1  1 -5 -1\r\n" );
document.write( "\r\n" );
document.write( "Nope. The remainder is -1 not 0, so 1 is not a solution.\r\n" );
document.write( "\r\n" );
document.write( "We try -1:\r\n" );
document.write( "\r\n" );
document.write( "-1 | 1  0 -6  4\r\n" );
document.write( "   |   -1  1  5\r\n" );
document.write( "     1 -1 -5  9\r\n" );
document.write( "\r\n" );
document.write( "Nope. The remainder is 9 not 0, so -1 is not a solution.\r\n" );
document.write( "\r\n" );
document.write( "We try 2:\r\n" );
document.write( "\r\n" );
document.write( " 2 | 1  0 -6  4\r\n" );
document.write( "   |    2  4 -4\r\n" );
document.write( "     1  2 -2  0\r\n" );
document.write( "\r\n" );
document.write( "Yep! The remainder is 0, so x=2 is a solution.  Therefore\r\n" );
document.write( "we have factored the left side as\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Using the zero-factor principle:\r\n" );
document.write( "\r\n" );
document.write( "Setting  of course gives the solution we just found,\r\n" );
document.write( "namely \r\n" );
document.write( "\r\n" );
document.write( "Setting \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "This does not factor, so we use the quadratic formula\r\n" );
document.write( "with , , \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( ", , and \r\n" );
document.write( "\r\n" );
document.write( "That's 3 real solutions.  No imaginary solutions.\r\n" );
document.write( "\r\n" );
document.write( "Edwin

\r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );