SOLUTION: 70s^2-165sr+90r^2 factor trinomial completely answer I had was 5 5(2a-3(7a-6r) but my answer was wrong please help Thank you

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: 70s^2-165sr+90r^2 factor trinomial completely answer I had was 5 5(2a-3(7a-6r) but my answer was wrong please help Thank you      Log On


   

Question 642394: 70s^2-165sr+90r^2 factor trinomial completely
answer I had was 5
5(2a-3(7a-6r) but my answer was wrong please help
Thank you
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
70s%5E2-165sr%2B90r%5E2 Start with the given expression.


5%2814s%5E2-33rs%2B18r%5E2%29 Factor out the GCF 5.


Now let's try to factor the inner expression 14s%5E2-33rs%2B18r%5E2

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Looking at the expression 14s%5E2-33rs%2B18r%5E2, we can see that the first coefficient is 14, the second coefficient is -33, and the last coefficient is 18.


Now multiply the first coefficient 14 by the last coefficient 18 to get %2814%29%2818%29=252.


Now the question is: what two whole numbers multiply to 252 (the previous product) and add to the second coefficient -33?


To find these two numbers, we need to list all of the factors of 252 (the previous product).


Factors of 252:
1,2,3,4,6,7,9,12,14,18,21,28,36,42,63,84,126,252
-1,-2,-3,-4,-6,-7,-9,-12,-14,-18,-21,-28,-36,-42,-63,-84,-126,-252


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


These factors pair up and multiply to 252.
1*252 = 252
2*126 = 252
3*84 = 252
4*63 = 252
6*42 = 252
7*36 = 252
9*28 = 252
12*21 = 252
14*18 = 252
(-1)*(-252) = 252
(-2)*(-126) = 252
(-3)*(-84) = 252
(-4)*(-63) = 252
(-6)*(-42) = 252
(-7)*(-36) = 252
(-9)*(-28) = 252
(-12)*(-21) = 252
(-14)*(-18) = 252

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -33:


First NumberSecond NumberSum
12521+252=253
21262+126=128
3843+84=87
4634+63=67
6426+42=48
7367+36=43
9289+28=37
122112+21=33
141814+18=32
-1-252-1+(-252)=-253
-2-126-2+(-126)=-128
-3-84-3+(-84)=-87
-4-63-4+(-63)=-67
-6-42-6+(-42)=-48
-7-36-7+(-36)=-43
-9-28-9+(-28)=-37
-12-21-12+(-21)=-33
-14-18-14+(-18)=-32



From the table, we can see that the two numbers -12 and -21 add to -33 (the middle coefficient).


So the two numbers -12 and -21 both multiply to 252 and add to -33


Now replace the middle term -33rs with -12rs-21rs. Remember, -12 and -21 add to -33. So this shows us that -12rs-21rs=-33rs.


14s%5E2%2Bhighlight%28-12rs-21rs%29%2B18r%5E2 Replace the second term -33rs with -12rs-21rs.


%2814s%5E2-12rs%29%2B%28-21rs%2B18r%5E2%29 Group the terms into two pairs.


2s%287s-6r%29%2B%28-21rs%2B18r%5E2%29 Factor out the GCF 2s from the first group.


2s%287s-6r%29-3r%287s-6r%29 Factor out 2s from the second group.


%282s-3r%29%287s-6r%29 Factor out 7s-6r

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So 14s%5E2-33rs%2B18r%5E2 factors to %282s-3r%29%287s-6r%29

this means 5%2814s%5E2-33rs%2B18r%5E2%29 factors to 5%282s-3r%29%287s-6r%29

So the final answer is 5%282s-3r%29%287s-6r%29