<PRE><B>Question 202586</B>

&lt;a name=&quot;top&quot;&gt;

&lt;a href=&quot;#ans&quot;&gt;Jump to Answer&lt;/a&gt;



Looking at {{{3x^2-17xy+24y^2}}} we can see that the first term is {{{3x^2}}} and the last term is {{{24y^2}}} where the coefficients are 3 and 24 respectively.


Now multiply the first coefficient 3 and the last coefficient 24 to get 72. Now what two numbers multiply to 72 and add to the  middle coefficient -17? Let&#39;s list all of the factors of 72:




Factors of 72:

1,2,3,4,6,8,9,12,18,24,36,72


-1,-2,-3,-4,-6,-8,-9,-12,-18,-24,-36,-72 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 72

1*72

2*36

3*24

4*18

6*12

8*9

(-1)*(-72)

(-2)*(-36)

(-3)*(-24)

(-4)*(-18)

(-6)*(-12)

(-8)*(-9)


note: remember two negative numbers multiplied together make a positive number



Now which of these pairs add to -17? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -17



&lt;table border=&quot;1&quot;&gt;&lt;th&gt;First Number&lt;/th&gt;&lt;th&gt;Second Number&lt;/th&gt;&lt;th&gt;Sum&lt;/th&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;72&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+72=73&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;36&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+36=38&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;24&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;3+24=27&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;18&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;4+18=22&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;6&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;12&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;6+12=18&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;8&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;9&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;8+9=17&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-72&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-72)=-73&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-36&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2+(-36)=-38&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-24&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3+(-24)=-27&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-18&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-4+(-18)=-22&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-6&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-12&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-6+(-12)=-18&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-8&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-9&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-8+(-9)=-17&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;





From this list we can see that -8 and -9 add up to -17 and multiply to 72



Now looking at the expression {{{3x^2-17xy+24y^2}}}, replace {{{-17xy}}} with {{{-8xy-9xy}}} (notice {{{-8xy-9xy}}} adds up to {{{-17xy}}}. So it is equivalent to {{{-17xy}}})


{{{3x^2+highlight(-8xy-9xy)+24y^2}}}



Now let&#39;s factor {{{3x^2-8xy-9xy+24y^2}}} by grouping:



{{{(3x^2-8xy)+(-9xy+24y^2)}}} Group like terms



{{{x(3x-8y)-3y(3x-8y)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{-3y}}} out of the second group



{{{(x-3y)(3x-8y)}}} Since we have a common term of {{{3x-8y}}}, we can combine like terms


So {{{3x^2-8xy-9xy+24y^2}}} factors to {{{(x-3y)(3x-8y)}}}



So this also means that {{{3x^2-17xy+24y^2}}} factors to {{{(x-3y)(3x-8y)}}} (since {{{3x^2-17xy+24y^2}}} is equivalent to {{{3x^2-8xy-9xy+24y^2}}})




------------------------------------------------------------



&lt;a name=&quot;ans&quot;&gt;

     Answer:

So {{{3x^2-17xy+24y^2}}} factors to {{{(x-3y)(3x-8y)}}}



In other words, {{{3x^2-17xy+24y^2=(x-3y)(3x-8y)}}}



&lt;a href=&quot;#top&quot;&gt;Jump to Top&lt;/a&gt;
</PRE>