<PRE><B>Question 711863</B>


Looking at the expression {{{x^2+9x+18}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{9}}}, and the last term is {{{18}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{18}}} to get {{{(1)(18)=18}}}.



Now the question is: what two whole numbers multiply to {{{18}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{9}}}?



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 {{{18}}} (the previous product).



Factors of {{{18}}}:

1,2,3,6,9,18

-1,-2,-3,-6,-9,-18



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{18}}}.

1*18 = 18
2*9 = 18
3*6 = 18
(-1)*(-18) = 18
(-2)*(-9) = 18
(-3)*(-6) = 18


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{9}}}:



&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;18&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+18=19&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;9&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+9=11&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;6&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;3+6=9&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;-18&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-18)=-19&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;-9&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2+(-9)=-11&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;-6&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3+(-6)=-9&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{3}}} and {{{6}}} add to {{{9}}} (the middle coefficient).



So the two numbers {{{3}}} and {{{6}}} both multiply to {{{18}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{9}}}



Now replace the middle term {{{9x}}} with {{{3x+6x}}}. Remember, {{{3}}} and {{{6}}} add to {{{9}}}. So this shows us that {{{3x+6x=9x}}}.



{{{x^2+highlight(3x+6x)+18}}} Replace the second term {{{9x}}} with {{{3x+6x}}}.



{{{(x^2+3x)+(6x+18)}}} Group the terms into two pairs.



{{{x(x+3)+(6x+18)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x+3)+6(x+3)}}} Factor out {{{6}}} 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.



{{{(x+6)(x+3)}}} Combine like terms. Or factor out the common term {{{x+3}}}



===============================================================



Answer:



So {{{x^2+9x+18}}} factors to {{{(x+6)(x+3)}}}.



In other words, {{{x^2+9x+18=(x+6)(x+3)}}}.



Note: you can check the answer by expanding {{{(x+6)(x+3)}}} to get {{{x^2+9x+18}}} or by graphing the original expression and the answer (the two graphs should be identical).
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