SOLUTION: Solve the following quadratic equation using factoring: 28x^2 + 30x + 32 = 0 Thank you for your help!

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Question 877860: Solve the following quadratic equation using factoring:
28x^2 + 30x + 32 = 0

Thank you for your help!
Found 2 solutions by Alan3354, richwmiller:
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
Solve the following quadratic equation using factoring:
28x2 + 30x + 32 = 0
Divide by 2

-------------
It's either
(x + p)*(14x + q) or
(2x + p)*(7x + q)
----------
p & q can be
1 & 16, 2 & 8, or 4 & 4
-------------
It's trial and error.
-------------
All the factors are +, makes it simpler.
15x is an odd number, that's a big hint.
Answer by richwmiller(17219)   (Show Source): You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Start with the given expression.



Factor out the GCF .



Now let's try to factor the inner expression



---------------------------------------------------------------



Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .



Now multiply the first coefficient by the last term to get .



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



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



Factors of :

1,2,4,7,8,14,16,28,32,56,112,224

-1,-2,-4,-7,-8,-14,-16,-28,-32,-56,-112,-224



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



These factors pair up and multiply to .

1*224 = 224
2*112 = 224
4*56 = 224
7*32 = 224
8*28 = 224
14*16 = 224
(-1)*(-224) = 224
(-2)*(-112) = 224
(-4)*(-56) = 224
(-7)*(-32) = 224
(-8)*(-28) = 224
(-14)*(-16) = 224


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



First NumberSecond NumberSum
12241+224=225
21122+112=114
4564+56=60
7327+32=39
8288+28=36
141614+16=30
-1-224-1+(-224)=-225
-2-112-2+(-112)=-114
-4-56-4+(-56)=-60
-7-32-7+(-32)=-39
-8-28-8+(-28)=-36
-14-16-14+(-16)=-30




From the table, we can see that there are no pairs of numbers which add to . So cannot be factored.



===============================================================





Answer:



So simply factors to



In other words, .
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