<PRE><B>Question 282083</B>


Looking at the expression {{{x^2-14x+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=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;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;/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 {{{-14x}}} with {{{-7x-7x}}}. Remember, {{{-7}}} and {{{-7}}} add to {{{-14}}}. So this shows us that {{{-7x-7x=-14x}}}.



{{{x^2+highlight(-7x-7x)+49}}} Replace the second term {{{-14x}}} with {{{-7x-7x}}}.



{{{(x^2-7x)+(-7x+49)}}} Group the terms into two pairs.



{{{x(x-7)+(-7x+49)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x-7)-7(x-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.



{{{(x-7)(x-7)}}} Combine like terms. Or factor out the common term {{{x-7}}}



{{{(x-7)^2}}} Condense the terms.



===============================================================



Answer:



So {{{x^2-14x+49}}} factors to {{{(x-7)^2}}}.



In other words, {{{x^2-14x+49=(x-7)^2}}}.



Note: you can check the answer by expanding {{{(x-7)^2}}} to get {{{x^2-14x+49}}} or by graphing the original expression and the answer (the two graphs should be identical).
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