<PRE><B>Question 581995</B>


Looking at the expression {{{s^2+14s+49}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{14}}}, and the last term is {{{49}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{49}}} to get {{{(1)(49)=49}}}.



Now the question is: what two whole numbers multiply to {{{49}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{14}}}?



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 {{{49}}} (the previous product).



Factors of {{{49}}}:

1,7,49

-1,-7,-49



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{49}}}.

1*49 = 49
7*7 = 49
(-1)*(-49) = 49
(-7)*(-7) = 49


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{14}}}:



&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;49&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+49=50&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;7&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;7&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;7+7=14&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;-49&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-49)=-50&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;-7&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-7+(-7)=-14&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{7}}} and {{{7}}} add to {{{14}}} (the middle coefficient).



So the two numbers {{{7}}} and {{{7}}} both multiply to {{{49}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{14}}}



Now replace the middle term {{{14s}}} with {{{7s+7s}}}. Remember, {{{7}}} and {{{7}}} add to {{{14}}}. So this shows us that {{{7s+7s=14s}}}.



{{{s^2+highlight(7s+7s)+49}}} Replace the second term {{{14s}}} with {{{7s+7s}}}.



{{{(s^2+7s)+(7s+49)}}} Group the terms into two pairs.



{{{s(s+7)+(7s+49)}}} Factor out the GCF {{{s}}} from the first group.



{{{s(s+7)+7(s+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.



{{{(s+7)(s+7)}}} Combine like terms. Or factor out the common term {{{s+7}}}



{{{(s+7)^2}}} Condense the terms.



===============================================================



Answer:



So {{{s^2+14s+49}}} factors to {{{(s+7)^2}}}.



In other words, {{{s^2+14s+49=(s+7)^2}}}.



Note: you can check the answer by expanding {{{(s+7)^2}}} to get {{{s^2+14s+49}}} or by graphing the original expression and the answer (the two graphs should be identical).
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