SOLUTION: factor x^2+36x+308

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Question 549824: factor x^2+36x+308
Found 2 solutions by jim_thompson5910, stanbon:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression x%5E2%2B36x%2B308, we can see that the first coefficient is 1, the second coefficient is 36, and the last term is 308.


Now multiply the first coefficient 1 by the last term 308 to get %281%29%28308%29=308.


Now the question is: what two whole numbers multiply to 308 (the previous product) and add to the second coefficient 36?


To find these two numbers, we need to list all of the factors of 308 (the previous product).


Factors of 308:
1,2,4,7,11,14,22,28,44,77,154,308
-1,-2,-4,-7,-11,-14,-22,-28,-44,-77,-154,-308


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


These factors pair up and multiply to 308.
1*308 = 308
2*154 = 308
4*77 = 308
7*44 = 308
11*28 = 308
14*22 = 308
(-1)*(-308) = 308
(-2)*(-154) = 308
(-4)*(-77) = 308
(-7)*(-44) = 308
(-11)*(-28) = 308
(-14)*(-22) = 308

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 36:


First NumberSecond NumberSum
13081+308=309
21542+154=156
4774+77=81
7447+44=51
112811+28=39
142214+22=36
-1-308-1+(-308)=-309
-2-154-2+(-154)=-156
-4-77-4+(-77)=-81
-7-44-7+(-44)=-51
-11-28-11+(-28)=-39
-14-22-14+(-22)=-36



From the table, we can see that the two numbers 14 and 22 add to 36 (the middle coefficient).


So the two numbers 14 and 22 both multiply to 308 and add to 36


Now replace the middle term 36x with 14x%2B22x. Remember, 14 and 22 add to 36. So this shows us that 14x%2B22x=36x.


x%5E2%2Bhighlight%2814x%2B22x%29%2B308 Replace the second term 36x with 14x%2B22x.


%28x%5E2%2B14x%29%2B%2822x%2B308%29 Group the terms into two pairs.


x%28x%2B14%29%2B%2822x%2B308%29 Factor out the GCF x from the first group.


x%28x%2B14%29%2B22%28x%2B14%29 Factor out 22 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%2B22%29%28x%2B14%29 Combine like terms. Or factor out the common term x%2B14


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Answer:


So x%5E2%2B36x%2B308 factors to %28x%2B22%29%28x%2B14%29.


In other words, x%5E2%2B36x%2B308=%28x%2B22%29%28x%2B14%29.


Note: you can check the answer by expanding %28x%2B22%29%28x%2B14%29 to get x%5E2%2B36x%2B308 or by graphing the original expression and the answer (the two graphs should be identical).

Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
factor x^2+36x+308
----
Quadratic Formula:
x = [-36 +- sqrt(36^2-4*308)]/2
-----
x = [-36 +- sqrt(-1231.97)]/2
-----
x = [-36 +- 35.1i]/2
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cheers,
Stan H.
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