SOLUTION: 16j^2 + 24j +9

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Question 223195: 16j^2 + 24j +9
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming you want to factor.


Looking at the expression 16j%5E2%2B24j%2B9, we can see that the first coefficient is 16, the second coefficient is 24, and the last term is 9.


Now multiply the first coefficient 16 by the last term 9 to get %2816%29%289%29=144.


Now the question is: what two whole numbers multiply to 144 (the previous product) and add to the second coefficient 24?


To find these two numbers, we need to list all of the factors of 144 (the previous product).


Factors of 144:
1,2,3,4,6,8,9,12,16,18,24,36,48,72,144
-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-36,-48,-72,-144


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


These factors pair up and multiply to 144.
1*144
2*72
3*48
4*36
6*24
8*18
9*16
12*12
(-1)*(-144)
(-2)*(-72)
(-3)*(-48)
(-4)*(-36)
(-6)*(-24)
(-8)*(-18)
(-9)*(-16)
(-12)*(-12)

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 24:


First NumberSecond NumberSum
11441+144=145
2722+72=74
3483+48=51
4364+36=40
6246+24=30
8188+18=26
9169+16=25
121212+12=24
-1-144-1+(-144)=-145
-2-72-2+(-72)=-74
-3-48-3+(-48)=-51
-4-36-4+(-36)=-40
-6-24-6+(-24)=-30
-8-18-8+(-18)=-26
-9-16-9+(-16)=-25
-12-12-12+(-12)=-24



From the table, we can see that the two numbers 12 and 12 add to 24 (the middle coefficient).


So the two numbers 12 and 12 both multiply to 144 and add to 24


Now replace the middle term 24j with 12j%2B12j. Remember, 12 and 12 add to 24. So this shows us that 12j%2B12j=24j.


16j%5E2%2Bhighlight%2812j%2B12j%29%2B9 Replace the second term 24j with 12j%2B12j.


%2816j%5E2%2B12j%29%2B%2812j%2B9%29 Group the terms into two pairs.


4j%284j%2B3%29%2B%2812j%2B9%29 Factor out the GCF 4j from the first group.


4j%284j%2B3%29%2B3%284j%2B3%29 Factor out 3 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.


%284j%2B3%29%284j%2B3%29 Combine like terms. Or factor out the common term 4j%2B3


%284j%2B3%29%5E2 Condense

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Answer:


So 16j%5E2%2B24j%2B9 factors to %284j%2B3%29%5E2.


Note: you can check the answer by FOILing %284j%2B3%29%5E2 to get 16j%5E2%2B24j%2B9 or by graphing the original expression and the answer (the two graphs should be identical).