<PRE><B>Question 694634</B>


{{{9x^2-36x+36}}} Start with the given expression.



{{{9(x^2-4x+4)}}} Factor out the GCF {{{9}}}.



Now let&#39;s try to factor the inner expression {{{x^2-4x+4}}}



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Looking at the expression {{{x^2-4x+4}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-4}}}, and the last term is {{{4}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{4}}} to get {{{(1)(4)=4}}}.



Now the question is: what two whole numbers multiply to {{{4}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{-4}}}?



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 {{{4}}} (the previous product).



Factors of {{{4}}}:

1,2,4

-1,-2,-4



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{4}}}.

1*4 = 4
2*2 = 4
(-1)*(-4) = 4
(-2)*(-2) = 4


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-4}}}:



&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;4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+4=5&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;2&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+2=4&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;-4&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-4)=-5&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-2&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-2&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-2+(-2)=-4&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-2}}} and {{{-2}}} add to {{{-4}}} (the middle coefficient).



So the two numbers {{{-2}}} and {{{-2}}} both multiply to {{{4}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-4}}}



Now replace the middle term {{{-4x}}} with {{{-2x-2x}}}. Remember, {{{-2}}} and {{{-2}}} add to {{{-4}}}. So this shows us that {{{-2x-2x=-4x}}}.



{{{x^2+highlight(-2x-2x)+4}}} Replace the second term {{{-4x}}} with {{{-2x-2x}}}.



{{{(x^2-2x)+(-2x+4)}}} Group the terms into two pairs.



{{{x(x-2)+(-2x+4)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x-2)-2(x-2)}}} Factor out {{{2}}} 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-2)(x-2)}}} Combine like terms. Or factor out the common term {{{x-2}}}



{{{(x-2)^2}}} Condense the terms.



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So {{{9(x^2-4x+4)}}} then factors further to {{{9(x-2)^2}}}



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Answer:



So {{{9x^2-36x+36}}} completely factors to {{{9(x-2)^2}}}.



In other words, {{{9x^2-36x+36=9(x-2)^2}}}.



Note: you can check the answer by expanding {{{9(x-2)^2}}} to get {{{9x^2-36x+36}}} or by graphing the original expression and the answer (the two graphs should be identical).
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