Question 183092
I assume that you want to factor.





Looking at {{{36x^2+12xy+y^2}}} we can see that the first term is {{{36x^2}}} and the last term is {{{y^2}}} where the coefficients are 36 and 1 respectively.


Now multiply the first coefficient 36 and the last coefficient 1 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 {{{36x^2+12xy+y^2}}}, replace {{{12xy}}} with {{{6xy+6xy}}} (notice {{{6xy+6xy}}} adds up to {{{12xy}}}. So it is equivalent to {{{12xy}}})


{{{36x^2+highlight(6xy+6xy)+y^2}}}



Now let's factor {{{36x^2+6xy+6xy+y^2}}} by grouping:



{{{(36x^2+6xy)+(6xy+y^2)}}} Group like terms



{{{6x(6x+y)+y(6x+y)}}} Factor out the GCF of {{{6x}}} out of the first group. Factor out the GCF of {{{y}}} out of the second group



{{{(6x+y)(6x+y)}}} Since we have a common term of {{{6x+y}}}, we can combine like terms


So {{{36x^2+6xy+6xy+y^2}}} factors to {{{(6x+y)(6x+y)}}}



So this also means that {{{36x^2+12xy+y^2}}} factors to {{{(6x+y)(6x+y)}}} (since {{{36x^2+12xy+y^2}}} is equivalent to {{{36x^2+6xy+6xy+y^2}}})



note:  {{{(6x+y)(6x+y)}}} is equivalent to  {{{(6x+y)^2}}} since the term {{{6x+y}}} occurs twice. So {{{36x^2+12xy+y^2}}} also factors to {{{(6x+y)^2}}}




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     Answer:

So {{{36x^2+12xy+y^2}}} factors to {{{(6x+y)^2}}}