<PRE><B>Question 452716</B>


Looking at the expression {{{x^2+6x+9}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{6}}}, and the last term is {{{9}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{9}}} to get {{{(1)(9)=9}}}.



Now the question is: what two whole numbers multiply to {{{9}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{6}}}?



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 {{{9}}} (the previous product).



Factors of {{{9}}}:

1,3,9

-1,-3,-9



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{9}}}.

1*9 = 9
3*3 = 9
(-1)*(-9) = 9
(-3)*(-3) = 9


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{6}}}:



&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;9&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+9=10&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;3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;3+3=6&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;-9&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-9)=-10&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;-3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3+(-3)=-6&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{3}}} and {{{3}}} add to {{{6}}} (the middle coefficient).



So the two numbers {{{3}}} and {{{3}}} both multiply to {{{9}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{6}}}



Now replace the middle term {{{6x}}} with {{{3x+3x}}}. Remember, {{{3}}} and {{{3}}} add to {{{6}}}. So this shows us that {{{3x+3x=6x}}}.



{{{x^2+highlight(3x+3x)+9}}} Replace the second term {{{6x}}} with {{{3x+3x}}}.



{{{(x^2+3x)+(3x+9)}}} Group the terms into two pairs.



{{{x(x+3)+(3x+9)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x+3)+3(x+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.



{{{(x+3)(x+3)}}} Combine like terms. Or factor out the common term {{{x+3}}}



{{{(x+3)^2}}} Condense the terms.



===============================================================



Answer:



So {{{x^2+6x+9}}} factors to {{{(x+3)^2}}}.



In other words, {{{x^2+6x+9=(x+3)^2}}}.



Note: you can check the answer by expanding {{{(x+3)^2}}} to get {{{x^2+6x+9}}} or by graphing the original expression and the answer (the two graphs should be identical).
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