SOLUTION: Factor {{{9s^2+16-24s}}}

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Question 131907: Factor 9s%5E2%2B16-24s
Answer by jim_thompson5910(35256) About Me  (Show Source):
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9s%5E2%2B16-24s Start with the given expression


9s%5E2-24s%2B16 Rearrange the terms


Looking at 9s%5E2-24s%2B16 we can see that the first term is 9s%5E2 and the last term is 16 where the coefficients are 9 and 16 respectively.

Now multiply the first coefficient 9 and the last coefficient 16 to get 144. Now what two numbers multiply to 144 and add to the middle coefficient -24? Let's list all of the factors of 144:



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

-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-36,-48,-72 ...List the negative factors as well. 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)

note: remember two negative numbers multiplied together make a positive number


Now which of these pairs add to -24? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -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 this list we can see that -12 and -12 add up to -24 and multiply to 144


Now looking at the expression 9s%5E2-24s%2B16, replace -24s with -12s%2B-12s (notice -12s%2B-12s adds up to -24s. So it is equivalent to -24s)

9s%5E2%2Bhighlight%28-12s%2B-12s%29%2B16


Now let's factor 9s%5E2-12s-12s%2B16 by grouping:


%289s%5E2-12s%29%2B%28-12s%2B16%29 Group like terms


3s%283s-4%29-4%283s-4%29 Factor out the GCF of 3s out of the first group. Factor out the GCF of -4 out of the second group


%283s-4%29%283s-4%29 Since we have a common term of 3s-4, we can combine like terms

So 9s%5E2-12s-12s%2B16 factors to %283s-4%29%283s-4%29


So this also means that 9s%5E2-24s%2B16 factors to %283s-4%29%283s-4%29 (since 9s%5E2-24s%2B16 is equivalent to 9s%5E2-12s-12s%2B16)


note: %283s-4%29%283s-4%29 is equivalent to %283s-4%29%5E2 since the term 3s-4 occurs twice. So 9s%5E2-24s%2B16 also factors to %283s-4%29%5E2



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Answer:
So 9s%5E2-24s%2B16 factors to %283s-4%29%5E2