document.write( "Question 282701: factor 4x^2 + 7xy - 15y^2 \n" ); document.write( "

Algebra.Com's Answer #205233 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "\n" ); document.write( "Looking at $\"4x%5E2%2B7xy-15y%5E2\"$ we can see that the first term is $\"4x%5E2\"$ and the last term is $\"-15y%5E2\"$ where the coefficients are 4 and -15 respectively.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 4 and the last coefficient -15 to get -60. Now what two numbers multiply to -60 and add to the middle coefficient 7? Let's list all of the factors of -60:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of -60:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,10,12,15,20,30,60\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60 ...List the negative factors as well. This will allow us to find all possible combinations\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to -60\r
\n" ); document.write( "\n" ); document.write( "(1)*(-60)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-30)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-20)\r
\n" ); document.write( "\n" ); document.write( "(4)*(-15)\r
\n" ); document.write( "\n" ); document.write( "(5)*(-12)\r
\n" ); document.write( "\n" ); document.write( "(6)*(-10)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(60)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(30)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(20)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(15)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(12)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(10)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "note: remember, the product of a negative and a positive number is a negative number\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to 7? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 7\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

First Number	Second Number	Sum
1	-60	1+(-60)=-59
2	-30	2+(-30)=-28
3	-20	3+(-20)=-17
4	-15	4+(-15)=-11
5	-12	5+(-12)=-7
6	-10	6+(-10)=-4
-1	60	-1+60=59
-2	30	-2+30=28
-3	20	-3+20=17
-4	15	-4+15=11
-5	12	-5+12=7
-6	10	-6+10=4

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From this list we can see that -5 and 12 add up to 7 and multiply to -60\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"4x%5E2%2B7xy-15y%5E2\"$ , replace $\"7xy\"$ with $\"-5xy%2B12xy\"$ (notice $\"-5xy%2B12xy\"$ adds up to $\"7xy\"$ . So it is equivalent to $\"7xy\"$ )\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"4x%5E2%2Bhighlight%28-5xy%2B12xy%29%2B-15y%5E2\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's factor $\"4x%5E2-5xy%2B12xy-15y%5E2\"$ by grouping:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%284x%5E2-5xy%29%2B%2812xy-15y%5E2%29\"$ Group like terms\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%284x-5y%29%2B3y%284x-5y%29\"$ Factor out the GCF of $\"x\"$ out of the first group. Factor out the GCF of $\"3y\"$ out of the second group\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x%2B3y%29%284x-5y%29\"$ Since we have a common term of $\"4x-5y\"$ , we can combine like terms\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"4x%5E2-5xy%2B12xy-15y%5E2\"$ factors to $\"%28x%2B3y%29%284x-5y%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So this also means that $\"4x%5E2%2B7xy-15y%5E2\"$ factors to $\"%28x%2B3y%29%284x-5y%29\"$ (since $\"4x%5E2%2B7xy-15y%5E2\"$ is equivalent to $\"4x%5E2-5xy%2B12xy-15y%5E2\"$ )\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "------------------------------------------------------------\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " Answer:\r
\n" ); document.write( "\n" ); document.write( "So $\"4x%5E2%2B7xy-15y%5E2\"$ factors to $\"%28x%2B3y%29%284x-5y%29\"$
\n" ); document.write( " \n" ); document.write( "

\n" );