document.write( "Question 110760: Hello can you answer these questions for me1\r
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\n" ); document.write( " 2y^2+8y+9y+36\r
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\n" ); document.write( " w^2-81v^2\r
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Algebra.Com's Answer #80781 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"x%5E2-3x-18\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-3\"$ , and the last term is $\"-18\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"-18\"$ to get $\"%281%29%28-18%29=-18\"$ .

Now the question is: what two whole numbers multiply to $\"-18\"$ (the previous product) and add to the second coefficient $\"-3\"$ ?

To find these two numbers, we need to list all of the factors of $\"-18\"$ (the previous product).

Factors of $\"-18\"$ :

1,2,3,6,9,18

-1,-2,-3,-6,-9,-18

Note: list the negative of each factor. This will allow us to find all possible combinations.

These factors pair up and multiply to $\"-18\"$ .

1*(-18) = -18
2*(-9) = -18
3*(-6) = -18
(-1)*(18) = -18
(-2)*(9) = -18
(-3)*(6) = -18

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-3\"$ :

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First Number	Second Number	Sum
1	-18	1+(-18)=-17
2	-9	2+(-9)=-7
3	-6	3+(-6)=-3
-1	18	-1+18=17
-2	9	-2+9=7
-3	6	-3+6=3

From the table, we can see that the two numbers $\"3\"$ and $\"-6\"$ add to $\"-3\"$ (the middle coefficient).

So the two numbers $\"3\"$ and $\"-6\"$ both multiply to $\"-18\"$ and add to $\"-3\"$

Now replace the middle term $\"-3x\"$ with $\"3x-6x\"$ . Remember, $\"3\"$ and $\"-6\"$ add to $\"-3\"$ . So this shows us that $\"3x-6x=-3x\"$ .

$\"x%5E2%2Bhighlight%283x-6x%29-18\"$ Replace the second term $\"-3x\"$ with $\"3x-6x\"$ .

$\"%28x%5E2%2B3x%29%2B%28-6x-18%29\"$ Group the terms into two pairs.

$\"x%28x%2B3%29%2B%28-6x-18%29\"$ Factor out the GCF $\"x\"$ from the first group.

$\"x%28x%2B3%29-6%28x%2B3%29\"$ Factor out $\"6\"$ from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.

$\"%28x-6%29%28x%2B3%29\"$ Combine like terms. Or factor out the common term $\"x%2B3\"$

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Answer:

So $\"x%5E2-3%2Ax-18\"$ factors to $\"%28x-6%29%28x%2B3%29\"$ .

In other words, $\"x%5E2-3%2Ax-18=%28x-6%29%28x%2B3%29\"$ .

Note: you can check the answer by expanding $\"%28x-6%29%28x%2B3%29\"$ to get $\"x%5E2-3%2Ax-18\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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\n" ); document.write( "\n" ); document.write( "\"Factor 2y^2+8y+9y+36\"\r
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\n" ); document.write( "\n" ); document.write( " $\"2y%5E2%2B8y%2B9y%2B36\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"%282y%5E2%2B8y%29%2B%289y%2B36%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"2y%28y%2B4%29%2B9%28y%2B4%29\"$ Factor out the GCF of $\"2y\"$ out of the first group. Factor out the GCF of $\"9\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%282y%2B9%29%28y%2B4%29\"$ Since we have a common term of $\"y%2B4\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"2y%5E2%2B8y%2B9y%2B36\"$ factors to $\"%282y%2B9%29%28y%2B4%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "\"Factor w^2-81v^2\"\r
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\n" ); document.write( "\n" ); document.write( " $\"w%5E2-81v%5E2\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( "Let $\"A%5E2=w%5E2\"$ and $\"B%5E2=81v%5E2\"$ . So we get this:\r
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\n" ); document.write( "\n" ); document.write( " $\"w%5E2-81v%5E2=A%5E2-B%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Since $\"A%5E2=w%5E2\"$ , A can be solved for:\r
\n" ); document.write( "\n" ); document.write( " $\"sqrt%28A%5E2%29=sqrt%28w%5E2%29\"$ Take the square root of both sides\r
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\n" ); document.write( "\n" ); document.write( " $\"A=w\"$ \r
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\n" ); document.write( "\n" ); document.write( "Since $\"B%5E2=81v%5E2\"$ , B can be solved for:\r
\n" ); document.write( "\n" ); document.write( " $\"sqrt%28B%5E2%29=sqrt%2881v%5E2%29\"$ Take the square root of both sides\r
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\n" ); document.write( "\n" ); document.write( " $\"B=9v\"$ \r
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\n" ); document.write( "\n" ); document.write( "Since we have a difference of squares, we can factor it like this:\r
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\n" ); document.write( "\n" ); document.write( " $\"A%5E2-B%5E2=%28A%2BB%29%28A-B%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace A and B\r
\n" ); document.write( "\n" ); document.write( " $\"w%5E2-81v%5E2=%28w%2B9v%29%28w-9v%29\"$ Plug in $\"A=w\"$ and $\"B=9v\"$ \r
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\n" ); document.write( "\n" ); document.write( "So the expression\r
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\n" ); document.write( "\n" ); document.write( " $\"w%5E2-81v%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%28w%2B9v%29%28w-9v%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Notice that if you foil the factored expression, you get the original expression. This verifies our answer.\r
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\n" ); document.write( "\n" ); document.write( "#4 \r
\n" ); document.write( "\n" ); document.write( "\"Factor 5x^2+23xy+12y^2\"\r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"5x%5E2%2B7xy-12y%5E2\"$ we can see that the first term is $\"5x%5E2\"$ and the last term is $\"-12y%5E2\"$ where the coefficients are 5 and -12 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 5 and the last coefficient -12 to get -60. Now what two numbers multiply to -60 and add to 7? Let's list all of the factors of -60:\r
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\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
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\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
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\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
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\n" ); document.write( "\n" ); document.write( "note: remember, the product of a negative and a positive number is a negative number\r
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\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
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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

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\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
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"5x%5E2%2B7xy-12y%5E2\"$ , replace $\"7xy\"$ with $\"-5xy%2B12xy\"$ (notice $\"-5xy%2B12xy\"$ adds up to $\"7xy\"$ . So it is equivalent to $\"7xy\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"5x%5E2%2Bhighlight%28-5xy%2B12xy%29%2B-12y%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"5x%5E2-5xy%2B12xy-12y%5E2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%285x%5E2-5xy%29%2B%2812xy-12y%5E2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"5x%28x-y%29%2B12y%28x-y%29\"$ Factor out the GCF of $\"5x\"$ out of the first group. Factor out the GCF of $\"12y\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%285x%2B12y%29%28x-y%29\"$ Since we have a common term of $\"x-y\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"5x%5E2-5xy%2B12xy-12y%5E2\"$ factors to $\"%285x%2B12y%29%28x-y%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"5x%5E2%2B7xy-12y%5E2\"$ factors to $\"%285x%2B12y%29%28x-y%29\"$ (since $\"5x%5E2%2B7xy-12y%5E2\"$ is equivalent to $\"5x%5E2-5xy%2B12xy-12y%5E2\"$ ) \n" ); document.write( "

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