document.write( "Question 195512: HOW DO I FACTOR PROBLEMS LIKE
\n" ); document.write( "X^2-9XY+14Y^2
\n" ); document.write( "OR
\n" ); document.write( "6a^2-5ab-4b^2
\n" ); document.write( "please show the work for it
\n" ); document.write( "thanks \n" ); document.write( "

Algebra.Com's Answer #146648 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
# 1\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Looking at $\"x%5E2-9xy%2B14y%5E2\"$ we can see that the first term is $\"x%5E2\"$ and the last term is $\"14y%5E2\"$ where the coefficients are 1 and 14 respectively.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient 14 to get 14. Now what two numbers multiply to 14 and add to the middle coefficient -9? Let's list all of the factors of 14:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of 14:\r
\n" ); document.write( "\n" ); document.write( "1,2,7,14\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "-1,-2,-7,-14 ...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 14\r
\n" ); document.write( "\n" ); document.write( "1*14\r
\n" ); document.write( "\n" ); document.write( "2*7\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-14)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-7)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "note: remember two negative numbers multiplied together make a positive number\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to -9? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -9\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

First Number	Second Number	Sum
1	14	1+14=15
2	7	2+7=9
-1	-14	-1+(-14)=-15
-2	-7	-2+(-7)=-9

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From this list we can see that -2 and -7 add up to -9 and multiply to 14\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"x%5E2-9xy%2B14y%5E2\"$ , replace $\"-9xy\"$ with $\"-2xy-7xy\"$ (notice $\"-2xy-7xy\"$ combines to $\"-9xy\"$ . So it is equivalent to $\"-9xy\"$ )\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%28-2xy-7xy%29%2B14y%5E2\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's factor $\"x%5E2-2xy-7xy%2B14y%5E2\"$ by grouping:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2-2xy%29%2B%28-7xy%2B14y%5E2%29\"$ Group like terms\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%28x-2y%29-7y%28x-2y%29\"$ Factor out the GCF of $\"x\"$ out of the first group. Factor out the GCF of $\"-7y\"$ out of the second group\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x-7y%29%28x-2y%29\"$ Since we have a common term of $\"x-2y\"$ , we can combine like terms\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"x%5E2-2xy-7xy%2B14y%5E2\"$ factors to $\"%28x-7y%29%28x-2y%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So this also means that $\"x%5E2-9xy%2B14y%5E2\"$ factors to $\"%28x-7y%29%28x-2y%29\"$ (since $\"x%5E2-9xy%2B14y%5E2\"$ is equivalent to $\"x%5E2-2xy-7xy%2B14y%5E2\"$ )\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 $\"x%5E2-9xy%2B14y%5E2\"$ factors to $\"%28x-7y%29%28x-2y%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "# 2\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Looking at $\"6a%5E2-5ab-4b%5E2\"$ we can see that the first term is $\"6a%5E2\"$ and the last term is $\"-4b%5E2\"$ where the coefficients are 6 and -4 respectively.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 6 and the last coefficient -4 to get -24. Now what two numbers multiply to -24 and add to the middle coefficient -5? Let's list all of the factors of -24:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of -24:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,12,24\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-12,-24 ...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 -24\r
\n" ); document.write( "\n" ); document.write( "(1)*(-24)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-12)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-8)\r
\n" ); document.write( "\n" ); document.write( "(4)*(-6)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(24)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(12)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(8)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(6)\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 -5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -5\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

First Number	Second Number	Sum
1	-24	1+(-24)=-23
2	-12	2+(-12)=-10
3	-8	3+(-8)=-5
4	-6	4+(-6)=-2
-1	24	-1+24=23
-2	12	-2+12=10
-3	8	-3+8=5
-4	6	-4+6=2

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From this list we can see that 3 and -8 add up to -5 and multiply to -24\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"6a%5E2-5ab-4b%5E2\"$ , replace $\"-5ab\"$ with $\"3ab%2B-8ab\"$ (notice $\"3ab%2B-8ab\"$ adds up to $\"-5ab\"$ . So it is equivalent to $\"-5ab\"$ )\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"6a%5E2%2Bhighlight%283ab%2B-8ab%29%2B-4b%5E2\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's factor $\"6a%5E2%2B3ab-8ab-4b%5E2\"$ by grouping:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%286a%5E2%2B3ab%29%2B%28-8ab-4b%5E2%29\"$ Group like terms\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"3a%282a%2Bb%29-4b%282a%2Bb%29\"$ Factor out the GCF of $\"3a\"$ out of the first group. Factor out the GCF of $\"-4b\"$ out of the second group\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%283a-4b%29%282a%2Bb%29\"$ Since we have a common term of $\"2a%2Bb\"$ , we can combine like terms\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"6a%5E2%2B3ab-8ab-4b%5E2\"$ factors to $\"%283a-4b%29%282a%2Bb%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So this also means that $\"6a%5E2-5ab-4b%5E2\"$ factors to $\"%283a-4b%29%282a%2Bb%29\"$ (since $\"6a%5E2-5ab-4b%5E2\"$ is equivalent to $\"6a%5E2%2B3ab-8ab-4b%5E2\"$ )\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 $\"6a%5E2-5ab-4b%5E2\"$ factors to $\"%283a-4b%29%282a%2Bb%29\"$
\n" ); document.write( " \n" ); document.write( "

\n" );