<PRE><B>Question 640408</B>


Looking at the expression {{{4x^2-20x+25}}}, we can see that the first coefficient is {{{4}}}, the second coefficient is {{{-20}}}, and the last term is {{{25}}}.



Now multiply the first coefficient {{{4}}} by the last term {{{25}}} to get {{{(4)(25)=100}}}.



Now the question is: what two whole numbers multiply to {{{100}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{-20}}}?



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 {{{100}}} (the previous product).



Factors of {{{100}}}:

1,2,4,5,10,20,25,50,100

-1,-2,-4,-5,-10,-20,-25,-50,-100



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{100}}}.

1*100 = 100
2*50 = 100
4*25 = 100
5*20 = 100
10*10 = 100
(-1)*(-100) = 100
(-2)*(-50) = 100
(-4)*(-25) = 100
(-5)*(-20) = 100
(-10)*(-10) = 100


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-20}}}:



&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;100&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+100=101&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;50&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+50=52&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;25&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;4+25=29&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;5&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;20&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;5+20=25&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;10&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;10&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;10+10=20&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;-100&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-100)=-101&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;-50&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2+(-50)=-52&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&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;-25&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-4+(-25)=-29&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-5&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-20&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-5+(-20)=-25&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-10&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-10&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-10+(-10)=-20&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-10}}} and {{{-10}}} add to {{{-20}}} (the middle coefficient).



So the two numbers {{{-10}}} and {{{-10}}} both multiply to {{{100}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-20}}}



Now replace the middle term {{{-20x}}} with {{{-10x-10x}}}. Remember, {{{-10}}} and {{{-10}}} add to {{{-20}}}. So this shows us that {{{-10x-10x=-20x}}}.



{{{4x^2+highlight(-10x-10x)+25}}} Replace the second term {{{-20x}}} with {{{-10x-10x}}}.



{{{(4x^2-10x)+(-10x+25)}}} Group the terms into two pairs.



{{{2x(2x-5)+(-10x+25)}}} Factor out the GCF {{{2x}}} from the first group.



{{{2x(2x-5)-5(2x-5)}}} Factor out {{{5}}} 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.



{{{(2x-5)(2x-5)}}} Combine like terms. Or factor out the common term {{{2x-5}}}



{{{(2x-5)^2}}} Condense the terms.



===============================================================



Answer:



So {{{4x^2-20x+25}}} factors to {{{(2x-5)^2}}}.



In other words, {{{4x^2-20x+25=(2x-5)^2}}}.



Note: you can check the answer by expanding {{{(2x-5)^2}}} to get {{{4x^2-20x+25}}} or by graphing the original expression and the answer (the two graphs should be identical).
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