Question 131907
{{{9s^2+16-24s}}} Start with the given expression



{{{9s^2-24s+16}}} Rearrange the terms



Looking at {{{9s^2-24s+16}}} we can see that the first term is {{{9s^2}}} 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


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">144</td><td>1+144=145</td></tr><tr><td align="center">2</td><td align="center">72</td><td>2+72=74</td></tr><tr><td align="center">3</td><td align="center">48</td><td>3+48=51</td></tr><tr><td align="center">4</td><td align="center">36</td><td>4+36=40</td></tr><tr><td align="center">6</td><td align="center">24</td><td>6+24=30</td></tr><tr><td align="center">8</td><td align="center">18</td><td>8+18=26</td></tr><tr><td align="center">9</td><td align="center">16</td><td>9+16=25</td></tr><tr><td align="center">12</td><td align="center">12</td><td>12+12=24</td></tr><tr><td align="center">-1</td><td align="center">-144</td><td>-1+(-144)=-145</td></tr><tr><td align="center">-2</td><td align="center">-72</td><td>-2+(-72)=-74</td></tr><tr><td align="center">-3</td><td align="center">-48</td><td>-3+(-48)=-51</td></tr><tr><td align="center">-4</td><td align="center">-36</td><td>-4+(-36)=-40</td></tr><tr><td align="center">-6</td><td align="center">-24</td><td>-6+(-24)=-30</td></tr><tr><td align="center">-8</td><td align="center">-18</td><td>-8+(-18)=-26</td></tr><tr><td align="center">-9</td><td align="center">-16</td><td>-9+(-16)=-25</td></tr><tr><td align="center">-12</td><td align="center">-12</td><td>-12+(-12)=-24</td></tr></table>



From this list we can see that -12 and -12 add up to -24 and multiply to 144



Now looking at the expression {{{9s^2-24s+16}}}, replace {{{-24s}}} with {{{-12s+-12s}}} (notice {{{-12s+-12s}}} adds up to {{{-24s}}}. So it is equivalent to {{{-24s}}})


{{{9s^2+highlight(-12s+-12s)+16}}}



Now let's factor {{{9s^2-12s-12s+16}}} by grouping:



{{{(9s^2-12s)+(-12s+16)}}} Group like terms



{{{3s(3s-4)-4(3s-4)}}} Factor out the GCF of {{{3s}}} out of the first group. Factor out the GCF of {{{-4}}} out of the second group



{{{(3s-4)(3s-4)}}} Since we have a common term of {{{3s-4}}}, we can combine like terms


So {{{9s^2-12s-12s+16}}} factors to {{{(3s-4)(3s-4)}}}



So this also means that {{{9s^2-24s+16}}} factors to {{{(3s-4)(3s-4)}}} (since {{{9s^2-24s+16}}} is equivalent to {{{9s^2-12s-12s+16}}})



note:  {{{(3s-4)(3s-4)}}} is equivalent to  {{{(3s-4)^2}}} since the term {{{3s-4}}} occurs twice. So {{{9s^2-24s+16}}} also factors to {{{(3s-4)^2}}}




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     Answer:

So {{{9s^2-24s+16}}} factors to {{{(3s-4)^2}}}