document.write( "Question 1113784: if a^2 + 1 = a, then the value of a^12 + a^6 +1 is \r
\n" ); document.write( "\n" ); document.write( "a)2\r
\n" ); document.write( "\n" ); document.write( "b)3\r
\n" ); document.write( "\n" ); document.write( "c)-3\r
\n" ); document.write( "\n" ); document.write( "d)1
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #728889 by ikleyn(52784) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "

\r\n" );
document.write( "If  a^2 + 1 = a,  then  a^2 - a + 1 = 0.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    Multiply both sides by (a+1). You will get   a^3 + 1 = 0,   or   a^3 = -1.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    So, the values of \"a\" that are the roots of the original equation, are the complex cubic roots of (-1).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "          // If you solve the original equation using the quadratic formula, you will get the same result.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    Therefore,  a^12 = (a^3)^4 = (-1)^4 = 1,   and\r\n" );
document.write( "\r\n" );
document.write( "                a^6  = (a^3)^2 = (-1)^2 = 1.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    Thus  a^12 + a^6 + 1 = 1 + 1 + 1 = 3.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The answer is 3.  Choice b).\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" );