Question 114249: Rewrite the middle term as the sum of two terms and then factor completely.
12w squared + 19w + 4
Found 3 solutions by stanbon, Fombitz, jim_thompson5910:
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! Rewrite the middle term as the sum of two terms and then factor completely.
12w squared + 19w + 4
-----------------
12w^2+16w+3w+4
=4w(3w+4)+(3w+4)
= (3w+4)(4w+1)
=================
Cheers,
Stan H.
Answer by Fombitz(32388)  (Show Source):
You can put this solution on YOUR website! 
Find a,b,c,d using the FOIL method (First, Outer, Inner, Last) and equating to the coefficients of your equation.
Now to expand the equation you use the FOIL method (First, Outer, Inner, Last)
First : 
Outer : 
Inner : 
Last : 
Comparing to your equation
1.
2.
3.
From 1, since ac=12, you can choose 12 and 1, 6 and 2, 4 and 3, and also 3 and 4, 2 and 6, and 1 and 12 for possible a-c combinations.
From 3, since bd=4, you can choose 4 and 1, 2 and 2, and 1 and 4 for possible b-d combinations.
Let's start with a=6 and c=2
From 2,
Notice the coefficients of b and d.
This solution set cannot be a solution because 2 even numbers will never sum to an odd number.
So 6 and 2 and 2 and 6 are out as possible a-c combinations.
Start again,
Let's choose a=12 and c=1,
From 2,
Now substitute possible b,d values.
 for b=4, d=1.
 for b=2, d=2.
 for b=1, d=4.
No solutions for a=12 and c=1.
Our last chance is 4 and 3.
Let's start with a=4 and c=3.
From 2,
Again, substitute possible b,d values.
 for b=4, d=1
 for b=2, d=2
 for b=1, d=4
Looks like a winner.
a=4,c=3,b=1,d=4
Answer by jim_thompson5910(35256) (Show Source):
You can put this solution on YOUR website!
| Solved by pluggable solver: Factoring using the AC method (Factor by Grouping) |
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 :
1,2,3,4,6,8,12,16,24,48
-1,-2,-3,-4,-6,-8,-12,-16,-24,-48
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to .
1*48 = 48
2*24 = 48
3*16 = 48
4*12 = 48
6*8 = 48
(-1)*(-48) = 48
(-2)*(-24) = 48
(-3)*(-16) = 48
(-4)*(-12) = 48
(-6)*(-8) = 48
Now let's add up each pair of factors to see if one pair adds to the middle coefficient :
| First Number | Second Number | Sum | | 1 | 48 | 1+48=49 | | 2 | 24 | 2+24=26 | | 3 | 16 | 3+16=19 | | 4 | 12 | 4+12=16 | | 6 | 8 | 6+8=14 | | -1 | -48 | -1+(-48)=-49 | | -2 | -24 | -2+(-24)=-26 | | -3 | -16 | -3+(-16)=-19 | | -4 | -12 | -4+(-12)=-16 | | -6 | -8 | -6+(-8)=-14 |
From the table, we can see that the two numbers  and  add to (the middle coefficient).
So the two numbers and both multiply to and add to
Now replace the middle term  with  . Remember, and add to . So this shows us that  .
 Replace the second term with .
 Group the terms into two pairs.
 Factor out the GCF  from the first group.
 Factor out 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.
 Combine like terms. Or factor out the common term 
===============================================================
Answer:
So  factors to .
In other words,  .
Note: you can check the answer by expanding to get or by graphing the original expression and the answer (the two graphs should be identical).
|
|
|
|
