document.write( "Question 275827: Solve for x.\r
\n" ); document.write( "\n" ); document.write( "(x+1)(x+3)(x+5)(x+7)=-15\r
\n" ); document.write( "\n" ); document.write( "I've worked this numerous times and get lost every time. Please explain how to find x. \n" ); document.write( "

Algebra.Com's Answer #854126 by ikleyn(53727) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Solve for x.
\n" ); document.write( "(x+1)(x+3)(x+5)(x+7)=-15
\n" ); document.write( "I've worked this numerous times and get lost every time. Please explain how to find x.
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " The post by @mananth does not contain and does not give a complete solution.\r
\n" ); document.write( "\n" ); document.write( " I came to do this job from the beginning to the end.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "Your starting equation is\r\n" );
document.write( "\r\n" );
document.write( "    (x+1)(x+3)(x+5)(x+7) = -15.    (1)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Although it leads to the fourth degree polynomial equation, it admits analytical solution\r\n" );
document.write( "not of transcendent level of complexity.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Let y = x+4.  Then\r\n" );
document.write( "\r\n" );
document.write( "    x+3 = y-1,  x+5 = y+1,  x+1 = y-3,  x+7 = y+3.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Therefore, equation (1) can be equivalently written in this form\r\n" );
document.write( "\r\n" );
document.write( "    (y-3)*(y-1)*(y+1)*(y+3 = -15.    (2)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Regrouping the terms, we get\r\n" );
document.write( "\r\n" );
document.write( "    (y^2-9)*(y^2-1) = -15,\r\n" );
document.write( "\r\n" );
document.write( "     y^4 - 9y^2 - y^2 + 9 = -15,\r\n" );
document.write( "\r\n" );
document.write( "     y^4 - 10y^2 +24 = 0,\r\n" );
document.write( "\r\n" );
document.write( "     (y^2-6)*(y^2-4) = 0.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Therefore, four routs of equation (2) are\r\n" );
document.write( "\r\n" );
document.write( "     = ,   = ,   = -2,   = 2.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "From here, the solutions to equation (1) are\r\n" );
document.write( "\r\n" );
document.write( "     = ,   = ,   = -2-4 = -6,   = 2-4 = -2.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "ANSWER.  The solutions to the given equation are  = ,   = ,   = -2-4 = -6,   = 2-4 = -2.\r\n" );
document.write( "

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

\n" );