<PRE><B>Question 383973</B>


Looking at the expression {{{y^2-4y+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 {{{-4y}}} with {{{-2y-2y}}}. Remember, {{{-2}}} and {{{-2}}} add to {{{-4}}}. So this shows us that {{{-2y-2y=-4y}}}.



{{{y^2+highlight(-2y-2y)+4}}} Replace the second term {{{-4y}}} with {{{-2y-2y}}}.



{{{(y^2-2y)+(-2y+4)}}} Group the terms into two pairs.



{{{y(y-2)+(-2y+4)}}} Factor out the GCF {{{y}}} from the first group.



{{{y(y-2)-2(y-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.



{{{(y-2)(y-2)}}} Combine like terms. Or factor out the common term {{{y-2}}}



{{{(y-2)^2}}} Condense the terms.



===============================================================



Answer:



So {{{y^2-4y+4}}} factors to {{{(y-2)^2}}}.



In other words, {{{y^2-4y+4=(y-2)^2}}}.



Note: you can check the answer by expanding {{{(y-2)^2}}} to get {{{y^2-4y+4}}} or by graphing the original expression and the answer (the two graphs should be identical).



If you need more help, email me at &lt;a href=&quot;mailto:jim_thompson5910@hotmail.com?Subject=Algebra%20Help&quot;&gt;jim_thompson5910@hotmail.com&lt;/a&gt;


Also, feel free to check out my &lt;a href=&quot;http://www.freewebs.com/jimthompson5910/home.html&quot;&gt;tutoring website&lt;/a&gt;


Jim</PRE>