<PRE><B>Question 183250</B>
I&#39;m assuming that you want to factor this.





Looking at the expression {{{w^2+14w+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
7*7
(-1)*(-49)
(-7)*(-7)


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 {{{14w}}} with {{{7w+7w}}}. Remember, {{{7}}} and {{{7}}} add to {{{14}}}. So this shows us that {{{7w+7w=14w}}}.



{{{w^2+highlight(7w+7w)+49}}} Replace the second term {{{14w}}} with {{{7w+7w}}}.



{{{(w^2+7w)+(7w+49)}}} Group the terms into two pairs.



{{{w(w+7)+(7w+49)}}} Factor out the GCF {{{w}}} from the first group.



{{{w(w+7)+7(w+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.



{{{(w+7)(w+7)}}} Combine like terms. Or factor out the common term {{{w+7}}}



{{{(w+7)^2}}} Condense the factors


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Answer:



So {{{w^2+14w+49}}} factors to {{{(w+7)^2}}}.



In other words, {{{w^2+14w+49=(w+7)^2}}}



Note: you can check the answer by FOILing {{{(w+7)(w+7)}}} to get {{{w^2+14w+49}}} or by graphing the original expression and the answer (the two graphs should be identical).
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