SOLUTION: I need so much help with solving quadratic equation by factoring! 1) 4x2 + 27x - 7=0

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Question 444490: I need so much help with solving quadratic equation by factoring!
1) 4x2 + 27x - 7=0
Answer by rwm(914) About Me  (Show Source):
You can put this solution on YOUR website!
4x^2 + 27x - 7=0
I will show you the process
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression 4x%5E2%2B27x-7, we can see that the first coefficient is 4, the second coefficient is 27, and the last term is -7.



Now multiply the first coefficient 4 by the last term -7 to get %284%29%28-7%29=-28.



Now the question is: what two whole numbers multiply to -28 (the previous product) and add to the second coefficient 27?



To find these two numbers, we need to list all of the factors of -28 (the previous product).



Factors of -28:

1,2,4,7,14,28

-1,-2,-4,-7,-14,-28



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



These factors pair up and multiply to -28.

1*(-28) = -28
2*(-14) = -28
4*(-7) = -28
(-1)*(28) = -28
(-2)*(14) = -28
(-4)*(7) = -28


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 27:



First NumberSecond NumberSum
1-281+(-28)=-27
2-142+(-14)=-12
4-74+(-7)=-3
-128-1+28=27
-214-2+14=12
-47-4+7=3




From the table, we can see that the two numbers -1 and 28 add to 27 (the middle coefficient).



So the two numbers -1 and 28 both multiply to -28 and add to 27



Now replace the middle term 27x with -x%2B28x. Remember, -1 and 28 add to 27. So this shows us that -x%2B28x=27x.



4x%5E2%2Bhighlight%28-x%2B28x%29-7 Replace the second term 27x with -x%2B28x.



%284x%5E2-x%29%2B%2828x-7%29 Group the terms into two pairs.



x%284x-1%29%2B%2828x-7%29 Factor out the GCF x from the first group.



x%284x-1%29%2B7%284x-1%29 Factor out 7 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%2B7%29%284x-1%29 Combine like terms. Or factor out the common term 4x-1



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Answer:



So 4%2Ax%5E2%2B27%2Ax-7 factors to %28x%2B7%29%284x-1%29.



In other words, 4%2Ax%5E2%2B27%2Ax-7=%28x%2B7%29%284x-1%29.



Note: you can check the answer by expanding %28x%2B7%29%284x-1%29 to get 4%2Ax%5E2%2B27%2Ax-7 or by graphing the original expression and the answer (the two graphs should be identical).