<PRE><B>Question 207723</B>

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Looking at the expression {{{9x^2+18xy-7y^2}}}, we can see that the first coefficient is {{{9}}}, the second coefficient is {{{18}}}, and the last coefficient is {{{-7}}}.



Now multiply the first coefficient {{{9}}} by the last coefficient {{{-7}}} to get {{{(9)(-7)=-63}}}.



Now the question is: what two whole numbers multiply to {{{-63}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{18}}}?



To find these two numbers, we need to list &lt;font size=4&gt;&lt;b&gt;all&lt;/b&gt;&lt;/font&gt; of the factors of {{{-63}}} (the previous product).



Factors of {{{-63}}}:

1,3,7,9,21,63

-1,-3,-7,-9,-21,-63



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{-63}}}.

1*(-63) = -63
3*(-21) = -63
7*(-9) = -63
(-1)*(63) = -63
(-3)*(21) = -63
(-7)*(9) = -63


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{18}}}:



&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;-63&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+(-63)=-62&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;-21&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;3+(-21)=-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;7&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;7+(-9)=-2&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;63&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+63=62&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;21&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-3+21=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;-7&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;-7+9=2&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-3}}} and {{{21}}} add to {{{18}}} (the middle coefficient).



So the two numbers {{{-3}}} and {{{21}}} both multiply to {{{-63}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{18}}}



Now replace the middle term {{{18xy}}} with {{{-3xy+21xy}}}. Remember, {{{-3}}} and {{{21}}} add to {{{18}}}. So this shows us that {{{-3xy+21xy=18xy}}}.



{{{9x^2+highlight(-3xy+21xy)-7y^2}}} Replace the second term {{{18xy}}} with {{{-3xy+21xy}}}.



{{{(9x^2-3xy)+(21xy-7y^2)}}} Group the terms into two pairs.



{{{3x(3x-y)+(21xy-7y^2)}}} Factor out the GCF {{{3x}}} from the first group.



{{{3x(3x-y)+7y(3x-y)}}} Factor out {{{7y}}} 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.



{{{(3x+7y)(3x-y)}}} Combine like terms. Or factor out the common term {{{3x-y}}}



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Answer:



So {{{9x^2+18xy-7y^2}}} factors to {{{(3x+7y)(3x-y)}}}.



In other words, {{{9x^2+18xy-7y^2=(3x+7y)(3x-y)}}}.



Note: you can check the answer by expanding {{{(3x+7y)(3x-y)}}} to get {{{9x^2+18xy-7y^2}}} or by graphing the original expression and the answer (the two graphs should be identical).


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