SOLUTION: 252v2+588vg+343g2

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Question 623409: 252v2+588vg+343g2
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

252v%5E2%2B588vg%2B343g%5E2 Start with the given expression.


7%2836v%5E2%2B84gv%2B49g%5E2%29 Factor out the GCF 7.


Now let's try to factor the inner expression 36v%5E2%2B84gv%2B49g%5E2


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Looking at the expression 36v%5E2%2B84gv%2B49g%5E2, we can see that the first coefficient is 36, the second coefficient is 84, and the last coefficient is 49.


Now multiply the first coefficient 36 by the last coefficient 49 to get %2836%29%2849%29=1764.


Now the question is: what two whole numbers multiply to 1764 (the previous product) and add to the second coefficient 84?


To find these two numbers, we need to list all of the factors of 1764 (the previous product).


Factors of 1764:
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
-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


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


These factors pair up and multiply to 1764.
1*1764 = 1764
2*882 = 1764
3*588 = 1764
4*441 = 1764
6*294 = 1764
7*252 = 1764
9*196 = 1764
12*147 = 1764
14*126 = 1764
18*98 = 1764
21*84 = 1764
28*63 = 1764
36*49 = 1764
42*42 = 1764
(-1)*(-1764) = 1764
(-2)*(-882) = 1764
(-3)*(-588) = 1764
(-4)*(-441) = 1764
(-6)*(-294) = 1764
(-7)*(-252) = 1764
(-9)*(-196) = 1764
(-12)*(-147) = 1764
(-14)*(-126) = 1764
(-18)*(-98) = 1764
(-21)*(-84) = 1764
(-28)*(-63) = 1764
(-36)*(-49) = 1764
(-42)*(-42) = 1764

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 84:


First NumberSecond NumberSum
117641+1764=1765
28822+882=884
35883+588=591
44414+441=445
62946+294=300
72527+252=259
91969+196=205
1214712+147=159
1412614+126=140
189818+98=116
218421+84=105
286328+63=91
364936+49=85
424242+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



From the table, we can see that the two numbers 42 and 42 add to 84 (the middle coefficient).


So the two numbers 42 and 42 both multiply to 1764 and add to 84


Now replace the middle term 84gv with 42gv%2B42gv. Remember, 42 and 42 add to 84. So this shows us that 42gv%2B42gv=84gv.


36v%5E2%2Bhighlight%2842gv%2B42gv%29%2B49g%5E2 Replace the second term 84gv with 42gv%2B42gv.


%2836v%5E2%2B42gv%29%2B%2842gv%2B49g%5E2%29 Group the terms into two pairs.


6v%286v%2B7g%29%2B%2842gv%2B49g%5E2%29 Factor out the GCF 6v from the first group.


6v%286v%2B7g%29%2B7g%286v%2B7g%29 Factor out 7g 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.


%286v%2B7g%29%286v%2B7g%29 Combine like terms. Or factor out the common term 6v%2B7g


%286v%2B7g%29%5E2 Condense the terms.


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So 7%2836v%5E2%2B84gv%2B49g%5E2%29 then factors further to 7%286v%2B7g%29%5E2


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Answer:


So 252v%5E2%2B588vg%2B343g%5E2 completely factors to 7%286v%2B7g%29%5E2.


In other words, 252v%5E2%2B588vg%2B343g%5E2=7%286v%2B7g%29%5E2.


Note: you can check the answer by expanding 7%286v%2B7g%29%5E2 to get 252v%5E2%2B588vg%2B343g%5E2 or by graphing the original expression and the answer (the two graphs should be identical).