SOLUTION: I'm stumped, can you help me answer this question? The zeros of the function f(x)=3x^2+4x-4 can be found by factoring as follows: (x+2)(3x-2) Is this True or False? Your help is

Algebra ->  Functions -> SOLUTION: I'm stumped, can you help me answer this question? The zeros of the function f(x)=3x^2+4x-4 can be found by factoring as follows: (x+2)(3x-2) Is this True or False? Your help is      Log On


   

Question 124267This question is from textbook Precalculus
: I'm stumped, can you help me answer this question?
The zeros of the function f(x)=3x^2+4x-4 can be found by factoring as follows: (x+2)(3x-2)
Is this True or False?
Your help is very well appreciated. Thank you so much. This question is from textbook Precalculus

Found 3 solutions by MathLover1, jim_thompson5910, josmiceli:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

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


Looking at the expression 3x%5E2%2B4x-4, we can see that the first coefficient is 3, the second coefficient is 4, and the last term is -4.



Now multiply the first coefficient 3 by the last term -4 to get %283%29%28-4%29=-12.



Now the question is: what two whole numbers multiply to -12 (the previous product) and add to the second coefficient 4?



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 4:



First NumberSecond NumberSum
1-121+(-12)=-11
2-62+(-6)=-4
3-43+(-4)=-1
-112-1+12=11
-26-2+6=4
-34-3+4=1




From the table, we can see that the two numbers -2 and 6 add to 4 (the middle coefficient).



So the two numbers -2 and 6 both multiply to -12 and add to 4



Now replace the middle term 4x with -2x%2B6x. Remember, -2 and 6 add to 4. So this shows us that -2x%2B6x=4x.



3x%5E2%2Bhighlight%28-2x%2B6x%29-4 Replace the second term 4x with -2x%2B6x.



%283x%5E2-2x%29%2B%286x-4%29 Group the terms into two pairs.



x%283x-2%29%2B%286x-4%29 Factor out the GCF x from the first group.



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



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



Answer:



So 3%2Ax%5E2%2B4%2Ax-4 factors to %28x%2B2%29%283x-2%29.



In other words, 3%2Ax%5E2%2B4%2Ax-4=%28x%2B2%29%283x-2%29.



Note: you can check the answer by expanding %28x%2B2%29%283x-2%29 to get 3%2Ax%5E2%2B4%2Ax-4 or by graphing the original expression and the answer (the two graphs should be identical).

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
This statement is true. Notice if we foil %28x%2B2%29%283x-2%29 we get

%28x%2B2%29%283x-2%29=3x%5E2-2x%2B6x-4=3x%5E2%2B4x-4

which is the original polynomial

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
%28x%2B2%29%283x-2%29=+0 yes. The zeros occur when f%28x%29+=+0
This is true when either x+=+-2 or x+=+2%2F3, so
these are the zeros