document.write( "Question 167668: factor completely
\n" ); document.write( "49x^2+84x+36 \n" ); document.write( "

Algebra.Com's Answer #123571 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"49x%5E2%2B84x%2B36\"$ , we can see that the first coefficient is $\"49\"$ , the second coefficient is $\"84\"$ , and the last term is $\"36\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"49\"$ by the last term $\"36\"$ to get $\"%2849%29%2836%29=1764\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"1764\"$ (the previous product) and add to the second coefficient $\"84\"$ ?\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"1764\"$ (the previous product).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"1764\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,7,9,12,14,18,21,28,36,42,49,63,84,98,126,147,196,252,294,441,588,882,1764\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-7,-9,-12,-14,-18,-21,-28,-36,-42,-49,-63,-84,-98,-126,-147,-196,-252,-294,-441,-588,-882,-1764\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. This will allow us to find all possible combinations.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to $\"1764\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*1764
\n" ); document.write( "2*882
\n" ); document.write( "3*588
\n" ); document.write( "4*441
\n" ); document.write( "6*294
\n" ); document.write( "7*252
\n" ); document.write( "9*196
\n" ); document.write( "12*147
\n" ); document.write( "14*126
\n" ); document.write( "18*98
\n" ); document.write( "21*84
\n" ); document.write( "28*63
\n" ); document.write( "36*49
\n" ); document.write( "42*42
\n" ); document.write( "(-1)*(-1764)
\n" ); document.write( "(-2)*(-882)
\n" ); document.write( "(-3)*(-588)
\n" ); document.write( "(-4)*(-441)
\n" ); document.write( "(-6)*(-294)
\n" ); document.write( "(-7)*(-252)
\n" ); document.write( "(-9)*(-196)
\n" ); document.write( "(-12)*(-147)
\n" ); document.write( "(-14)*(-126)
\n" ); document.write( "(-18)*(-98)
\n" ); document.write( "(-21)*(-84)
\n" ); document.write( "(-28)*(-63)
\n" ); document.write( "(-36)*(-49)
\n" ); document.write( "(-42)*(-42)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"84\"$ :\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

First Number	Second Number	Sum
1	1764	1+1764=1765
2	882	2+882=884
3	588	3+588=591
4	441	4+441=445
6	294	6+294=300
7	252	7+252=259
9	196	9+196=205
12	147	12+147=159
14	126	14+126=140
18	98	18+98=116
21	84	21+84=105
28	63	28+63=91
36	49	36+49=85
42	42	42+42=84
-1	-1764	-1+(-1764)=-1765
-2	-882	-2+(-882)=-884
-3	-588	-3+(-588)=-591
-4	-441	-4+(-441)=-445
-6	-294	-6+(-294)=-300
-7	-252	-7+(-252)=-259
-9	-196	-9+(-196)=-205
-12	-147	-12+(-147)=-159
-14	-126	-14+(-126)=-140
-18	-98	-18+(-98)=-116
-21	-84	-21+(-84)=-105
-28	-63	-28+(-63)=-91
-36	-49	-36+(-49)=-85
-42	-42	-42+(-42)=-84

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"42\"$ and $\"42\"$ add to $\"84\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"42\"$ and $\"42\"$ both multiply to $\"1764\"$ and add to $\"84\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"84x\"$ with $\"42x%2B42x\"$ . Remember, $\"42\"$ and $\"42\"$ add to $\"84\"$ . So this shows us that $\"42x%2B42x=84x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"49x%5E2%2Bhighlight%2842x%2B42x%29%2B36\"$ Replace the second term $\"84x\"$ with $\"42x%2B42x\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%2849x%5E2%2B42x%29%2B%2842x%2B36%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"7x%287x%2B6%29%2B%2842x%2B36%29\"$ Factor out the GCF $\"7x\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"7x%287x%2B6%29%2B6%287x%2B6%29\"$ 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.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%287x%2B6%29%287x%2B6%29\"$ Combine like terms. Or factor out the common term $\"7x%2B6\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%287x%2B6%29%5E2\"$ Simplify\r
\n" ); document.write( "\n" ); document.write( "---------------------------------------------\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Answer:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"49x%5E2%2B84x%2B36\"$ factors to $\"%287x%2B6%29%5E2\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%287x%2B6%29%5E2\"$ to get $\"49x%5E2%2B84x%2B36\"$ or by graphing the original expression and the answer (the two graphs should be identical).
\n" ); document.write( " \n" ); document.write( "

\n" );