document.write( "Question 1130128: factor\r
\n" ); document.write( "\n" ); document.write( "5a^2-7ab-6b^2 \n" ); document.write( "
Algebra.Com's Answer #746777 by ikleyn(52832)\"\" \"About 
You can put this solution on YOUR website!
.
\n" ); document.write( "
\r\n" );
document.write( "Consider an equation\r\n" );
document.write( "\r\n" );
document.write( "5a^2 - 7ab - 6b^2 = 0.       (1)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Divide both sides by b. You will get the equation\r\n" );
document.write( "\r\n" );
document.write( "\"5%28a%2Fb%29%5E2+-+7%28a%2Fb%29+-+6\" = 0.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Introduce new variable  x = \"a%2Fb\".  Then the last equation takes the form\r\n" );
document.write( "\r\n" );
document.write( "5x^2 - 7x - 6 = 0.    (2)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Find its roots using the quadratic formula\r\n" );
document.write( "\r\n" );
document.write( "\"x%5B1%2C2%5D\" = \"%287+%2B-+sqrt%287%5E2+-4%2A5%28-6%29%29%29%2F%282%2A5%29\" = \"%287+%2B-+sqrt%28169%29%29%2F10\" = \"%287+%2B-+13%29%2F10\".\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, the roots are  \"-6%2F10\" = \"-3%2F5\"  and  2.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It means that the polynomial (2) can be factored in this way\r\n" );
document.write( "\r\n" );
document.write( "5x^2 - 7x - 6  = 5*(x-2)*(x+3/5) = (x-2)*(5x+3).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Returning to variables \"a\" and \"b\", you get\r\n" );
document.write( "\r\n" );
document.write( "5a^2 - 7ab - 6b^2 = (a-2b)*(5a+3b).       ANSWER\r\n" );
document.write( "
\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The lesson to learn from the solution is THIS :\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "    If you have difficulties factoring such a quadratic polynomial of 2 variables \r\n" );
document.write( "        (in other words, if you do not see the factoring formula immediately before your eyes in your mind)\r\n" );
document.write( "     do the following (as I did in my solution)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "        - reduce your quadratic polynomial of two variable to the quadratic polynomial of one new variable;\r\n" );
document.write( "\r\n" );
document.write( "        - then find the roots of the reduced quadratic polynomial using the quadratic formula;\r\n" );
document.write( "\r\n" );
document.write( "        - then factor the reduced quadratic polynomial using its roots;\r\n" );
document.write( "\r\n" );
document.write( "        - then return back to the original variables \"a\" and \"b\".\r\n" );
document.write( "
\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "This method works ALWAYS like an army tank.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "It means that  if the factoring formula does exist over rational (or integer) numbers, \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "                        then the method provides / (guarantees) you will find the formula.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );