Question 527944


Looking at the expression {{{2x^2-5xy-18y^2}}}, we can see that the first coefficient is {{{2}}}, the second coefficient is {{{-5}}}, and the last coefficient is {{{-18}}}.



Now multiply the first coefficient {{{2}}} by the last coefficient {{{-18}}} to get {{{(2)(-18)=-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 {{{-5}}}?



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 {{{-5}}}:



<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)=-35</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)=-16</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)=-9</font></td></tr><tr><td  align="center"><font color=red>4</font></td><td  align="center"><font color=red>-9</font></td><td  align="center"><font color=red>4+(-9)=-5</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)=0</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=35</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=16</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=9</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=5</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=0</font></td></tr></table>



From the table, we can see that the two numbers {{{4}}} and {{{-9}}} add to {{{-5}}} (the middle coefficient).



So the two numbers {{{4}}} and {{{-9}}} both multiply to {{{-36}}} <font size=4><b>and</b></font> add to {{{-5}}}



Now replace the middle term {{{-5xy}}} with {{{4xy-9xy}}}. Remember, {{{4}}} and {{{-9}}} add to {{{-5}}}. So this shows us that {{{4xy-9xy=-5xy}}}.



{{{2x^2+highlight(4xy-9xy)-18y^2}}} Replace the second term {{{-5xy}}} with {{{4xy-9xy}}}.



{{{(2x^2+4xy)+(-9xy-18y^2)}}} Group the terms into two pairs.



{{{2x(x+2y)+(-9xy-18y^2)}}} Factor out the GCF {{{2x}}} from the first group.



{{{2x(x+2y)-9y(x+2y)}}} Factor out {{{-9y}}} from the second group.



{{{(2x-9y)(x+2y)}}} Factor out the GCF {{{x+2y}}} from the entire expression.




So {{{2x^2-5xy-18y^2}}} completely factors to {{{(2x-9y)(x+2y)}}}



In other words, {{{2x^2-5xy-18y^2=(2x-9y)(x+2y)}}}



If you need more help, email me at <a href="mailto:jim_thompson5910@hotmail.com">jim_thompson5910@hotmail.com</a>


Also, please consider visiting my website: <a href="http://www.freewebs.com/jimthompson5910/home.html">http://www.freewebs.com/jimthompson5910/home.html</a> and making a donation. Thank you


Jim