Question 119708


Looking at {{{x^2-12x+36}}} we can see that the first term is {{{x^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 {{{x^2-12x+36}}}, replace {{{-12x}}} with {{{-6x+-6x}}} (notice {{{-6x+-6x}}} adds up to {{{-12x}}}. So it is equivalent to {{{-12x}}})


{{{x^2+highlight(-6x+-6x)+36}}}



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



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



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



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


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



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



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




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     Answer:

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