Question 405579
Rearrange the terms to get {{{t^2-25t+144}}}





Looking at {{{t^2-25t+144}}} we can see that the first term is {{{t^2}}} and the last term is {{{144}}} where the coefficients are 1 and 144 respectively.


Now multiply the first coefficient 1 and the last coefficient 144 to get 144. Now what two numbers multiply to 144 and add to the  middle coefficient -25? 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 -25? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -25


<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 -9 and -16 add up to -25 and multiply to 144



Now looking at the expression {{{t^2-25t+144}}}, replace {{{-25t}}} with {{{-9t-16t}}} (notice {{{-9t-16t}}} adds up to {{{-25t}}}. So it is equivalent to {{{-25t}}})


{{{t^2+highlight(-9t-16t)+144}}}



Now let's factor {{{t^2-9t-16t+144}}} by grouping:



{{{(t^2-9t)+(-16t+144)}}} Group like terms



{{{t(t-9)-16(t-9)}}} Factor out the GCF of {{{t}}} out of the first group. Factor out the GCF of {{{-16}}} out of the second group



{{{(t-16)(t-9)}}} Since we have a common term of {{{t-9}}}, we can combine like terms


So {{{t^2-9t-16t+144}}} factors to {{{(t-16)(t-9)}}}



So this also means that {{{t^2-25t+144}}} factors to {{{(t-16)(t-9)}}} (since {{{t^2-25t+144}}} is equivalent to {{{t^2-9t-16t+144}}})




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     Answer:

So {{{t^2-25t+144}}} factors to {{{(t-16)(t-9)}}}



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