<PRE><B>Question 634761</B>


Looking at the expression {{{49m^2-42m+9}}}, we can see that the first coefficient is {{{49}}}, the second coefficient is {{{-42}}}, and the last term is {{{9}}}.



Now multiply the first coefficient {{{49}}} by the last term {{{9}}} to get {{{(49)(9)=441}}}.



Now the question is: what two whole numbers multiply to {{{441}}} (the previous product) &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to the second coefficient {{{-42}}}?



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 {{{441}}} (the previous product).



Factors of {{{441}}}:

1,3,7,9,21,49,63,147,441

-1,-3,-7,-9,-21,-49,-63,-147,-441



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to {{{441}}}.

1*441 = 441
3*147 = 441
7*63 = 441
9*49 = 441
21*21 = 441
(-1)*(-441) = 441
(-3)*(-147) = 441
(-7)*(-63) = 441
(-9)*(-49) = 441
(-21)*(-21) = 441


Now let&#39;s add up each pair of factors to see if one pair adds to the middle coefficient {{{-42}}}:



&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;441&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;1+441=442&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;147&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;3+147=150&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;63&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;7+63=70&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;9&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;9+49=58&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;21&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;21&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;21+21=42&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;-441&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-1+(-441)=-442&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-147&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-3+(-147)=-150&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;-63&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-7+(-63)=-70&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=black&gt;-9&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;-9+(-49)=-58&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;tr&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-21&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-21&lt;/font&gt;&lt;/td&gt;&lt;td  align=&quot;center&quot;&gt;&lt;font color=red&gt;-21+(-21)=-42&lt;/font&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;



From the table, we can see that the two numbers {{{-21}}} and {{{-21}}} add to {{{-42}}} (the middle coefficient).



So the two numbers {{{-21}}} and {{{-21}}} both multiply to {{{441}}} &lt;font size=4&gt;&lt;b&gt;and&lt;/b&gt;&lt;/font&gt; add to {{{-42}}}



Now replace the middle term {{{-42m}}} with {{{-21m-21m}}}. Remember, {{{-21}}} and {{{-21}}} add to {{{-42}}}. So this shows us that {{{-21m-21m=-42m}}}.



{{{49m^2+highlight(-21m-21m)+9}}} Replace the second term {{{-42m}}} with {{{-21m-21m}}}.



{{{(49m^2-21m)+(-21m+9)}}} Group the terms into two pairs.



{{{7m(7m-3)+(-21m+9)}}} Factor out the GCF {{{7m}}} from the first group.



{{{7m(7m-3)-3(7m-3)}}} Factor out {{{3}}} 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.



{{{(7m-3)(7m-3)}}} Combine like terms. Or factor out the common term {{{7m-3}}}



{{{(7m-3)^2}}} Condense the terms.



===============================================================



Answer:



So {{{49m^2-42m+9}}} factors to {{{(7m-3)^2}}}.



In other words, {{{49m^2-42m+9=(7m-3)^2}}}.



Note: you can check the answer by expanding {{{(7m-3)^2}}} to get {{{49m^2-42m+9}}} or by graphing the original expression and the answer (the two graphs should be identical).


&lt;font color=&quot;red&quot;&gt;--------------------------------------------------------------------------------------------------------------&lt;/font&gt;
If you need more help, email me at &lt;a href=&quot;mailto:jim_thompson5910@hotmail.com?Subject=I%20Need%20Algebra%20Help&quot;&gt;jim_thompson5910@hotmail.com&lt;/a&gt;


Also, please consider visiting my website: &lt;a href=&quot;http://www.freewebs.com/jimthompson5910/home.html&quot;&gt;http://www.freewebs.com/jimthompson5910/home.html&lt;/a&gt; and making a donation. Thank you


Jim
&lt;font color=&quot;red&quot;&gt;--------------------------------------------------------------------------------------------------------------&lt;/font&gt;</PRE>