SOLUTION: I need help solving 6x^2+14x+4

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Question 257548: I need help solving 6x^2+14x+4



Found 2 solutions by nerdybill, richwmiller:
Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
6x%5E2%2B14x%2B4
.
First factor out what is common among all terms (2):
2%283x%5E2%2B7x%2B2%29
.
2%283x%2B1%29%28x%2B2%29
.

Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
You almost had it right
^ is power
^2 is squared
^3 cubed
6x^2+14x+4
This expression can't really be solved because it is not an equation.
There is no equal sign. But we can factor it.
If the expression were set equal to 0 then the two factors could be set to 0 and solved.
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


6%2Ax%5E2%2B14%2Ax%2B4 Start with the given expression.



2%283x%5E2%2B7x%2B2%29 Factor out the GCF 2.



Now let's try to factor the inner expression 3x%5E2%2B7x%2B2



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Looking at the expression 3x%5E2%2B7x%2B2, we can see that the first coefficient is 3, the second coefficient is 7, and the last term is 2.



Now multiply the first coefficient 3 by the last term 2 to get %283%29%282%29=6.



Now the question is: what two whole numbers multiply to 6 (the previous product) and add to the second coefficient 7?



To find these two numbers, we need to list all of the factors of 6 (the previous product).



Factors of 6:

1,2,3,6

-1,-2,-3,-6



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



These factors pair up and multiply to 6.

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


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



First NumberSecond NumberSum
161+6=7
232+3=5
-1-6-1+(-6)=-7
-2-3-2+(-3)=-5




From the table, we can see that the two numbers 1 and 6 add to 7 (the middle coefficient).



So the two numbers 1 and 6 both multiply to 6 and add to 7



Now replace the middle term 7x with x%2B6x. Remember, 1 and 6 add to 7. So this shows us that x%2B6x=7x.



3x%5E2%2Bhighlight%28x%2B6x%29%2B2 Replace the second term 7x with x%2B6x.



%283x%5E2%2Bx%29%2B%286x%2B2%29 Group the terms into two pairs.



x%283x%2B1%29%2B%286x%2B2%29 Factor out the GCF x from the first group.



x%283x%2B1%29%2B2%283x%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%283x%2B1%29 Combine like terms. Or factor out the common term 3x%2B1



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So 2%283x%5E2%2B7x%2B2%29 then factors further to 2%28x%2B2%29%283x%2B1%29



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Answer:



So 6%2Ax%5E2%2B14%2Ax%2B4 completely factors to 2%28x%2B2%29%283x%2B1%29.



In other words, 6%2Ax%5E2%2B14%2Ax%2B4=2%28x%2B2%29%283x%2B1%29.



Note: you can check the answer by expanding 2%28x%2B2%29%283x%2B1%29 to get 6%2Ax%5E2%2B14%2Ax%2B4 or by graphing the original expression and the answer (the two graphs should be identical).