Question 319182


{{{27x^2-36xy+12y^2}}} Start with the given expression



{{{3(9x^2-12xy+4y^2)}}} Factor out the GCF {{{3}}}



Now let's focus on the inner expression {{{9x^2-12xy+4y^2}}}





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


Now multiply the first coefficient 9 and the last coefficient 4 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 {{{9x^2-12xy+4y^2}}}, replace {{{-12xy}}} with {{{-6xy+-6xy}}} (notice {{{-6xy+-6xy}}} adds up to {{{-12xy}}}. So it is equivalent to {{{-12xy}}})


{{{9x^2+highlight(-6xy+-6xy)+4y^2}}}



Now let's factor {{{9x^2-6xy-6xy+4y^2}}} by grouping:



{{{(9x^2-6xy)+(-6xy+4y^2)}}} Group like terms



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



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


So {{{9x^2-6xy-6xy+4y^2}}} factors to {{{(3x-2y)(3x-2y)}}}



So this also means that {{{9x^2-12xy+4y^2}}} factors to {{{(3x-2y)(3x-2y)}}} (since {{{9x^2-12xy+4y^2}}} is equivalent to {{{9x^2-6xy-6xy+4y^2}}})



note:  {{{(3x-2y)(3x-2y)}}} is equivalent to  {{{(3x-2y)^2}}} since the term {{{3x-2y}}} occurs twice. So {{{9x^2-12xy+4y^2}}} also factors to {{{(3x-2y)^2}}}




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So our expression goes from {{{3(9x^2-12xy+4y^2)}}} and factors further to {{{3(3x-2y)^2}}}



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Answer:


So {{{27x^2-36xy+12y^2}}} factors to {{{3(3x-2y)^2}}}