```Question 634761

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) <font size=4><b>and</b></font> add to the second coefficient {{{-42}}}?

To find these two numbers, we need to list <font size=4><b>all</b></font> 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's add up each pair of factors to see if one pair adds to the middle coefficient {{{-42}}}:

<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td  align="center"><font color=black>1</font></td><td  align="center"><font color=black>441</font></td><td  align="center"><font color=black>1+441=442</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>147</font></td><td  align="center"><font color=black>3+147=150</font></td></tr><tr><td  align="center"><font color=black>7</font></td><td  align="center"><font color=black>63</font></td><td  align="center"><font color=black>7+63=70</font></td></tr><tr><td  align="center"><font color=black>9</font></td><td  align="center"><font color=black>49</font></td><td  align="center"><font color=black>9+49=58</font></td></tr><tr><td  align="center"><font color=black>21</font></td><td  align="center"><font color=black>21</font></td><td  align="center"><font color=black>21+21=42</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>-441</font></td><td  align="center"><font color=black>-1+(-441)=-442</font></td></tr><tr><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>-147</font></td><td  align="center"><font color=black>-3+(-147)=-150</font></td></tr><tr><td  align="center"><font color=black>-7</font></td><td  align="center"><font color=black>-63</font></td><td  align="center"><font color=black>-7+(-63)=-70</font></td></tr><tr><td  align="center"><font color=black>-9</font></td><td  align="center"><font color=black>-49</font></td><td  align="center"><font color=black>-9+(-49)=-58</font></td></tr><tr><td  align="center"><font color=red>-21</font></td><td  align="center"><font color=red>-21</font></td><td  align="center"><font color=red>-21+(-21)=-42</font></td></tr></table>

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}}} <font size=4><b>and</b></font> 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.

===============================================================

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).

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