Question 704620


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



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



Now let's try to factor the inner expression {{{4x^2-12x+9}}}



---------------------------------------------------------------



Looking at the expression {{{4x^2-12x+9}}}, we can see that the first coefficient is {{{4}}}, the second coefficient is {{{-12}}}, and the last term is {{{9}}}.



Now multiply the first coefficient {{{4}}} by the last term {{{9}}} to get {{{(4)(9)=36}}}.



Now the question is: what two whole numbers multiply to {{{36}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-12}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{36}}} (the previous product).



Factors of {{{36}}}:

1,2,3,4,6,9,12,18,36

-1,-2,-3,-4,-6,-9,-12,-18,-36



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{36}}}.

1*36 = 36
2*18 = 36
3*12 = 36
4*9 = 36
6*6 = 36
(-1)*(-36) = 36
(-2)*(-18) = 36
(-3)*(-12) = 36
(-4)*(-9) = 36
(-6)*(-6) = 36


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-12}}}:



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



From the table, we can see that the two numbers {{{-6}}} and {{{-6}}} add to {{{-12}}} (the middle coefficient).



So the two numbers {{{-6}}} and {{{-6}}} both multiply to {{{36}}} <font size=4><b>and</b></font> add to {{{-12}}}



Now replace the middle term {{{-12x}}} with {{{-6x-6x}}}. Remember, {{{-6}}} and {{{-6}}} add to {{{-12}}}. So this shows us that {{{-6x-6x=-12x}}}.



{{{4x^2+highlight(-6x-6x)+9}}} Replace the second term {{{-12x}}} with {{{-6x-6x}}}.



{{{(4x^2-6x)+(-6x+9)}}} Group the terms into two pairs.



{{{2x(2x-3)+(-6x+9)}}} Factor out the GCF {{{2x}}} from the first group.



{{{2x(2x-3)-3(2x-3)}}} Factor out {{{3}}} from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.



{{{(2x-3)(2x-3)}}} Combine like terms. Or factor out the common term {{{2x-3}}}



{{{(2x-3)^2}}} Condense the terms.



--------------------------------------------------



So {{{3(4x^2-12x+9)}}} then factors further to {{{3(2x-3)^2}}}



===============================================================



Answer:



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



In other words, {{{12x^2-36x+27=3(2x-3)^2}}}.



Note: you can check the answer by expanding {{{3(2x-3)^2}}} to get {{{12x^2-36x+27}}} or by graphing the original expression and the answer (the two graphs should be identical).


<font color="red">--------------------------------------------------------------------------------------------------------------</font>


If you need more help, you can email me at <a href="mailto:jim_thompson5910@hotmail.com?Subject=I%20Need%20Algebra%20Help">jim_thompson5910@hotmail.com</a>


To find other ways to contact me, you can also visit my website: <a href="http://www.freewebs.com/jimthompson5910/home.html">http://www.freewebs.com/jimthompson5910/home.html</a>


Thanks,


Jim


<font color="red">--------------------------------------------------------------------------------------------------------------</font>