SOLUTION: question 1. a^2-18a-88=0 question 2. u^2+7u+12=0

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Question 888806: question 1. a^2-18a-88=0
question 2. u^2+7u+12=0
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression a%5E2-18a-88, we can see that the first coefficient is 1, the second coefficient is -18, and the last term is -88.



Now multiply the first coefficient 1 by the last term -88 to get %281%29%28-88%29=-88.



Now the question is: what two whole numbers multiply to -88 (the previous product) and add to the second coefficient -18?



To find these two numbers, we need to list all of the factors of -88 (the previous product).



Factors of -88:

1,2,4,8,11,22,44,88

-1,-2,-4,-8,-11,-22,-44,-88



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



These factors pair up and multiply to -88.

1*(-88) = -88
2*(-44) = -88
4*(-22) = -88
8*(-11) = -88
(-1)*(88) = -88
(-2)*(44) = -88
(-4)*(22) = -88
(-8)*(11) = -88


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -18:



First NumberSecond NumberSum
1-881+(-88)=-87
2-442+(-44)=-42
4-224+(-22)=-18
8-118+(-11)=-3
-188-1+88=87
-244-2+44=42
-422-4+22=18
-811-8+11=3




From the table, we can see that the two numbers 4 and -22 add to -18 (the middle coefficient).



So the two numbers 4 and -22 both multiply to -88 and add to -18



Now replace the middle term -18a with 4a-22a. Remember, 4 and -22 add to -18. So this shows us that 4a-22a=-18a.



a%5E2%2Bhighlight%284a-22a%29-88 Replace the second term -18a with 4a-22a.



%28a%5E2%2B4a%29%2B%28-22a-88%29 Group the terms into two pairs.



a%28a%2B4%29%2B%28-22a-88%29 Factor out the GCF a from the first group.



a%28a%2B4%29-22%28a%2B4%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.



%28a-22%29%28a%2B4%29 Combine like terms. Or factor out the common term a%2B4



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Answer:



So a%5E2-18%2Aa-88 factors to %28a-22%29%28a%2B4%29.



In other words, a%5E2-18%2Aa-88=%28a-22%29%28a%2B4%29.



Note: you can check the answer by expanding %28a-22%29%28a%2B4%29 to get a%5E2-18%2Aa-88 or by graphing the original expression and the answer (the two graphs should be identical).


Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression u%5E2%2B7u%2B12, we can see that the first coefficient is 1, the second coefficient is 7, and the last term is 12.



Now multiply the first coefficient 1 by the last term 12 to get %281%29%2812%29=12.



Now the question is: what two whole numbers multiply to 12 (the previous product) and add to the second coefficient 7?



To find these two numbers, we need to list all of the factors of 12 (the previous product).



Factors of 12:

1,2,3,4,6,12

-1,-2,-3,-4,-6,-12



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



These factors pair up and multiply to 12.

1*12 = 12
2*6 = 12
3*4 = 12
(-1)*(-12) = 12
(-2)*(-6) = 12
(-3)*(-4) = 12


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 7:



First NumberSecond NumberSum
1121+12=13
262+6=8
343+4=7
-1-12-1+(-12)=-13
-2-6-2+(-6)=-8
-3-4-3+(-4)=-7




From the table, we can see that the two numbers 3 and 4 add to 7 (the middle coefficient).



So the two numbers 3 and 4 both multiply to 12 and add to 7



Now replace the middle term 7u with 3u%2B4u. Remember, 3 and 4 add to 7. So this shows us that 3u%2B4u=7u.



u%5E2%2Bhighlight%283u%2B4u%29%2B12 Replace the second term 7u with 3u%2B4u.



%28u%5E2%2B3u%29%2B%284u%2B12%29 Group the terms into two pairs.



u%28u%2B3%29%2B%284u%2B12%29 Factor out the GCF u from the first group.



u%28u%2B3%29%2B4%28u%2B3%29 Factor out 4 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.



%28u%2B4%29%28u%2B3%29 Combine like terms. Or factor out the common term u%2B3



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Answer:



So u%5E2%2B7%2Au%2B12 factors to %28u%2B4%29%28u%2B3%29.



In other words, u%5E2%2B7%2Au%2B12=%28u%2B4%29%28u%2B3%29.



Note: you can check the answer by expanding %28u%2B4%29%28u%2B3%29 to get u%5E2%2B7%2Au%2B12 or by graphing the original expression and the answer (the two graphs should be identical).