<PRE><B>Question 738916</B>


Looking at the expression {{{4n^2-28n+49}}}, we can see that the first coefficient is {{{4}}}, the second coefficient is {{{-28}}}, and the last term is {{{49}}}.



Now multiply the first coefficient {{{4}}} by the last term {{{49}}} to get {{{(4)(49)=196}}}.



Now the question is: what two whole numbers multiply to {{{196}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{-28}}}?



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 {{{196}}} (the previous product).



Factors of {{{196}}}:

1,2,4,7,14,28,49,98,196

-1,-2,-4,-7,-14,-28,-49,-98,-196



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{196}}}.

1*196 = 196
2*98 = 196
4*49 = 196
7*28 = 196
14*14 = 196
(-1)*(-196) = 196
(-2)*(-98) = 196
(-4)*(-49) = 196
(-7)*(-28) = 196
(-14)*(-14) = 196


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-28}}}:



&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;196&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+196=197&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;98&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;2+98=100&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;49&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;4+49=53&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;7&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;28&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;7+28=35&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;14&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;14&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;14+14=28&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;-196&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-196)=-197&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;-98&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-2+(-98)=-100&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;-49&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-4+(-49)=-53&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-7&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-28&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-7+(-28)=-35&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-14&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-14&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-14+(-14)=-28&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-14}}} and {{{-14}}} add to {{{-28}}} (the middle coefficient).



So the two numbers {{{-14}}} and {{{-14}}} both multiply to {{{196}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-28}}}



Now replace the middle term {{{-28n}}} with {{{-14n-14n}}}. Remember, {{{-14}}} and {{{-14}}} add to {{{-28}}}. So this shows us that {{{-14n-14n=-28n}}}.



{{{4n^2+highlight(-14n-14n)+49}}} Replace the second term {{{-28n}}} with {{{-14n-14n}}}.



{{{(4n^2-14n)+(-14n+49)}}} Group the terms into two pairs.



{{{2n(2n-7)+(-14n+49)}}} Factor out the GCF {{{2n}}} from the first group.



{{{2n(2n-7)-7(2n-7)}}} Factor out {{{7}}} 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.



{{{(2n-7)(2n-7)}}} Combine like terms. Or factor out the common term {{{2n-7}}}



{{{(2n-7)^2}}} Condense the terms.



===============================================================



Answer:



So {{{4n^2-28n+49}}} factors to {{{(2n-7)^2}}}.



In other words, {{{4n^2-28n+49=(2n-7)^2}}}.



Note: you can check the answer by expanding {{{(2n-7)^2}}} to get {{{4n^2-28n+49}}} or by graphing the original expression and the answer (the two graphs should be identical).
</PRE>