SOLUTION: How do you factor a trinomial 14x^2+35x+14.

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Question 260982: How do you factor a trinomial 14x^2+35x+14.
Found 3 solutions by richwmiller, Greenfinch, checkley77:
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)


14%2Ax%5E2%2B35%2Ax%2B14 Start with the given expression.



7%282x%5E2%2B5x%2B2%29 Factor out the GCF 7.



Now let's try to factor the inner expression 2x%5E2%2B5x%2B2



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Looking at the expression 2x%5E2%2B5x%2B2, we can see that the first coefficient is 2, the second coefficient is 5, and the last term is 2.



Now multiply the first coefficient 2 by the last term 2 to get %282%29%282%29=4.



Now the question is: what two whole numbers multiply to 4 (the previous product) and add to the second coefficient 5?



To find these two numbers, we need to list all of the factors of 4 (the previous product).



Factors of 4:

1,2,4

-1,-2,-4



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



These factors pair up and multiply to 4.

1*4 = 4
2*2 = 4
(-1)*(-4) = 4
(-2)*(-2) = 4


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 5:



First NumberSecond NumberSum
141+4=5
222+2=4
-1-4-1+(-4)=-5
-2-2-2+(-2)=-4




From the table, we can see that the two numbers 1 and 4 add to 5 (the middle coefficient).



So the two numbers 1 and 4 both multiply to 4 and add to 5



Now replace the middle term 5x with x%2B4x. Remember, 1 and 4 add to 5. So this shows us that x%2B4x=5x.



2x%5E2%2Bhighlight%28x%2B4x%29%2B2 Replace the second term 5x with x%2B4x.



%282x%5E2%2Bx%29%2B%284x%2B2%29 Group the terms into two pairs.



x%282x%2B1%29%2B%284x%2B2%29 Factor out the GCF x from the first group.



x%282x%2B1%29%2B2%282x%2B1%29 Factor out 2 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%2B2%29%282x%2B1%29 Combine like terms. Or factor out the common term 2x%2B1



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So 7%282x%5E2%2B5x%2B2%29 then factors further to 7%28x%2B2%29%282x%2B1%29



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Answer:



So 14%2Ax%5E2%2B35%2Ax%2B14 completely factors to 7%28x%2B2%29%282x%2B1%29.



In other words, 14%2Ax%5E2%2B35%2Ax%2B14=7%28x%2B2%29%282x%2B1%29.



Note: you can check the answer by expanding 7%28x%2B2%29%282x%2B1%29 to get 14%2Ax%5E2%2B35%2Ax%2B14 or by graphing the original expression and the answer (the two graphs should be identical).

Answer by Greenfinch(383) About Me  (Show Source):
You can put this solution on YOUR website!
First there is a factor of 7 so
7(2x^2 + 5 x +2) which factorizes to
7(2x + 1 )(x + 2 )

Answer by checkley77(12844) About Me  (Show Source):
You can put this solution on YOUR website!
14x^2+35x+14
The factors are :
(14x+1)(1x+14)=196x+1x=197x
(14x+14)(1x+1)=14x+14x=28x
(2x+7)(7x+2)=4x+49x=53x
(2x+2)(7x+7)=14x+14x=28x
(14x+2)(x+7)=98+2x=99x
(14x+7)(x+2)=28x+7x=35x ans.