Question 146261


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


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


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



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



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



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



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


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



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



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




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     Answer:

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