Question 118297
{{{4t^2-12t+9-w^2}}} Start with the given expression


Let's focus on the polynomial {{{4t^2-12t+9}}}





Looking at {{{4t^2-12t+9}}} we can see that the first term is {{{4t^2}}} and the last term is {{{9}}} where the coefficients are 4 and 9 respectively.


Now multiply the first coefficient 4 and the last coefficient 9 to get 36. Now what two numbers multiply to 36 and add to the  middle coefficient -12? Let's list all of the factors of 36:




Factors of 36:

1,2,3,4,6,9,12,18


-1,-2,-3,-4,-6,-9,-12,-18 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 36

1*36

2*18

3*12

4*9

6*6

(-1)*(-36)

(-2)*(-18)

(-3)*(-12)

(-4)*(-9)

(-6)*(-6)


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



Now which of these pairs add to -12? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -12


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">36</td><td>1+36=37</td></tr><tr><td align="center">2</td><td align="center">18</td><td>2+18=20</td></tr><tr><td align="center">3</td><td align="center">12</td><td>3+12=15</td></tr><tr><td align="center">4</td><td align="center">9</td><td>4+9=13</td></tr><tr><td align="center">6</td><td align="center">6</td><td>6+6=12</td></tr><tr><td align="center">-1</td><td align="center">-36</td><td>-1+(-36)=-37</td></tr><tr><td align="center">-2</td><td align="center">-18</td><td>-2+(-18)=-20</td></tr><tr><td align="center">-3</td><td align="center">-12</td><td>-3+(-12)=-15</td></tr><tr><td align="center">-4</td><td align="center">-9</td><td>-4+(-9)=-13</td></tr><tr><td align="center">-6</td><td align="center">-6</td><td>-6+(-6)=-12</td></tr></table>



From this list we can see that -6 and -6 add up to -12 and multiply to 36



Now looking at the expression {{{4t^2-12t+9}}}, replace {{{-12t}}} with {{{-6t+-6t}}} (notice {{{-6t+-6t}}} adds up to {{{-12t}}}. So it is equivalent to {{{-12t}}})


{{{4t^2+highlight(-6t+-6t)+9}}}



Now let's factor {{{4t^2-6t-6t+9}}} by grouping:



{{{(4t^2-6t)+(-6t+9)}}} Group like terms



{{{2t(2t-3)-3(2t-3)}}} Factor out the GCF of {{{2t}}} out of the first group. Factor out the GCF of {{{-3}}} out of the second group



{{{(2t-3)(2t-3)}}} Since we have a common term of {{{2t-3}}}, we can combine like terms



So {{{4t^2-12t+9}}} factors to {{{(2t-3)(2t-3)}}} which is {{{(2t-3)^2}}}



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So {{{4t^2-12t+9-w^2}}} becomes {{{(2t-3)^2-w^2}}}



Notice how we have a difference of squares. This means we can use the difference of squares formula to factor further.


Let {{{A=2t-3}}} and {{{B=w}}}


{{{A^2-B^2}}} So we then get this



{{{A^2-B^2=(A+B)(A-B)}}} Now factor using the difference of squares



{{{(2t-3+w)(2t-3-w)}}} Now replace A with {{{2t-3}}} and B with {{{w}}}





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Answer:


So {{{4t^2-12t+9-w^2}}} completely factors to {{{(2t-3+w)(2t-3-w)}}}