document.write( "Question 1209229: The real numbers a and b satisfy a - b = 2 and a^3 - b^3 = 8.\r
\n" ); document.write( "\n" ); document.write( "(a) Find all possible values of ab.
\n" ); document.write( "(b) Find all possible values of a + b.
\n" ); document.write( "(c) Find all possible values of a and b. \n" ); document.write( "

Algebra.Com's Answer #847871 by ikleyn(52781) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "The real numbers a and b satisfy a - b = 2 and a^3 - b^3 = 8.\r
\n" ); document.write( "\n" ); document.write( "(a) Find all possible values of ab.
\n" ); document.write( "(b) Find all possible values of a + b.
\n" ); document.write( "(c) Find all possible values of a and b.
\n" ); document.write( "~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "Factor a^3 - b^3\r\n" );
document.write( "\r\n" );
document.write( "    a^3 - b^3 = (a-b)*(a^2 + ab + b^2).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Then from the second equation \r\n" );
document.write( "\r\n" );
document.write( "    (a-b)*(a^2 + ab + b^2) = 8.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Replace here a-b by 2, based on the first equation.  You will get\r\n" );
document.write( "\r\n" );
document.write( "    a^2 + ab + b^2 = 4.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Substitute here b = a-2.  You will get\r\n" );
document.write( "\r\n" );
document.write( "    a^2 + a*(a-2) + (a-2)^2 = 4,\r\n" );
document.write( "\r\n" );
document.write( "    a^2 + a^2 - 2a + a^2 - 4a + 4 = 4,\r\n" );
document.write( "\r\n" );
document.write( "    3a^2 - 6a = 0,\r\n" );
document.write( "\r\n" );
document.write( "    3a(a-2) = 0.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It gives two roots for \"a\" :  a = 0  and  a = 2.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "If  a= 0,  then  b = a-2 = -2.\r\n" );
document.write( "\r\n" );
document.write( "If  a= 2,  then  b = a-2 =  0.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, the solutions for the given system of equations are these pairs  (a,b) = (0,-2)  and  (a,b) = (2,0).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Having this, you can compute everything what you want / (you need) and answer all the questions.\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" );