```Question 376650

{{{49s^2-84s+36}}} Rearrange the terms.

Looking at the expression {{{49s^2-84s+36}}}, we can see that the first coefficient is {{{49}}}, the second coefficient is {{{-84}}}, and the last term is {{{36}}}.

Now multiply the first coefficient {{{49}}} by the last term {{{36}}} to get {{{(49)(36)=1764}}}.

Now the question is: what two whole numbers multiply to {{{1764}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-84}}}?

To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{1764}}} (the previous product).

Factors of {{{1764}}}:

1,2,3,4,6,7,9,12,14,18,21,28,36,42,49,63,84,98,126,147,196,252,294,441,588,882,1764

-1,-2,-3,-4,-6,-7,-9,-12,-14,-18,-21,-28,-36,-42,-49,-63,-84,-98,-126,-147,-196,-252,-294,-441,-588,-882,-1764

Note: list the negative of each factor. This will allow us to find all possible combinations.

These factors pair up and multiply to {{{1764}}}.

1*1764 = 1764
2*882 = 1764
3*588 = 1764
4*441 = 1764
6*294 = 1764
7*252 = 1764
9*196 = 1764
12*147 = 1764
14*126 = 1764
18*98 = 1764
21*84 = 1764
28*63 = 1764
36*49 = 1764
42*42 = 1764
(-1)*(-1764) = 1764
(-2)*(-882) = 1764
(-3)*(-588) = 1764
(-4)*(-441) = 1764
(-6)*(-294) = 1764
(-7)*(-252) = 1764
(-9)*(-196) = 1764
(-12)*(-147) = 1764
(-14)*(-126) = 1764
(-18)*(-98) = 1764
(-21)*(-84) = 1764
(-28)*(-63) = 1764
(-36)*(-49) = 1764
(-42)*(-42) = 1764

Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-84}}}:

<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>1764</font></td><td  align="center"><font color=black>1+1764=1765</font></td></tr><tr><td  align="center"><font color=black>2</font></td><td  align="center"><font color=black>882</font></td><td  align="center"><font color=black>2+882=884</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>588</font></td><td  align="center"><font color=black>3+588=591</font></td></tr><tr><td  align="center"><font color=black>4</font></td><td  align="center"><font color=black>441</font></td><td  align="center"><font color=black>4+441=445</font></td></tr><tr><td  align="center"><font color=black>6</font></td><td  align="center"><font color=black>294</font></td><td  align="center"><font color=black>6+294=300</font></td></tr><tr><td  align="center"><font color=black>7</font></td><td  align="center"><font color=black>252</font></td><td  align="center"><font color=black>7+252=259</font></td></tr><tr><td  align="center"><font color=black>9</font></td><td  align="center"><font color=black>196</font></td><td  align="center"><font color=black>9+196=205</font></td></tr><tr><td  align="center"><font color=black>12</font></td><td  align="center"><font color=black>147</font></td><td  align="center"><font color=black>12+147=159</font></td></tr><tr><td  align="center"><font color=black>14</font></td><td  align="center"><font color=black>126</font></td><td  align="center"><font color=black>14+126=140</font></td></tr><tr><td  align="center"><font color=black>18</font></td><td  align="center"><font color=black>98</font></td><td  align="center"><font color=black>18+98=116</font></td></tr><tr><td  align="center"><font color=black>21</font></td><td  align="center"><font color=black>84</font></td><td  align="center"><font color=black>21+84=105</font></td></tr><tr><td  align="center"><font color=black>28</font></td><td  align="center"><font color=black>63</font></td><td  align="center"><font color=black>28+63=91</font></td></tr><tr><td  align="center"><font color=black>36</font></td><td  align="center"><font color=black>49</font></td><td  align="center"><font color=black>36+49=85</font></td></tr><tr><td  align="center"><font color=black>42</font></td><td  align="center"><font color=black>42</font></td><td  align="center"><font color=black>42+42=84</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>-1764</font></td><td  align="center"><font color=black>-1+(-1764)=-1765</font></td></tr><tr><td  align="center"><font color=black>-2</font></td><td  align="center"><font color=black>-882</font></td><td  align="center"><font color=black>-2+(-882)=-884</font></td></tr><tr><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>-588</font></td><td  align="center"><font color=black>-3+(-588)=-591</font></td></tr><tr><td  align="center"><font color=black>-4</font></td><td  align="center"><font color=black>-441</font></td><td  align="center"><font color=black>-4+(-441)=-445</font></td></tr><tr><td  align="center"><font color=black>-6</font></td><td  align="center"><font color=black>-294</font></td><td  align="center"><font color=black>-6+(-294)=-300</font></td></tr><tr><td  align="center"><font color=black>-7</font></td><td  align="center"><font color=black>-252</font></td><td  align="center"><font color=black>-7+(-252)=-259</font></td></tr><tr><td  align="center"><font color=black>-9</font></td><td  align="center"><font color=black>-196</font></td><td  align="center"><font color=black>-9+(-196)=-205</font></td></tr><tr><td  align="center"><font color=black>-12</font></td><td  align="center"><font color=black>-147</font></td><td  align="center"><font color=black>-12+(-147)=-159</font></td></tr><tr><td  align="center"><font color=black>-14</font></td><td  align="center"><font color=black>-126</font></td><td  align="center"><font color=black>-14+(-126)=-140</font></td></tr><tr><td  align="center"><font color=black>-18</font></td><td  align="center"><font color=black>-98</font></td><td  align="center"><font color=black>-18+(-98)=-116</font></td></tr><tr><td  align="center"><font color=black>-21</font></td><td  align="center"><font color=black>-84</font></td><td  align="center"><font color=black>-21+(-84)=-105</font></td></tr><tr><td  align="center"><font color=black>-28</font></td><td  align="center"><font color=black>-63</font></td><td  align="center"><font color=black>-28+(-63)=-91</font></td></tr><tr><td  align="center"><font color=black>-36</font></td><td  align="center"><font color=black>-49</font></td><td  align="center"><font color=black>-36+(-49)=-85</font></td></tr><tr><td  align="center"><font color=red>-42</font></td><td  align="center"><font color=red>-42</font></td><td  align="center"><font color=red>-42+(-42)=-84</font></td></tr></table>

From the table, we can see that the two numbers {{{-42}}} and {{{-42}}} add to {{{-84}}} (the middle coefficient).

So the two numbers {{{-42}}} and {{{-42}}} both multiply to {{{1764}}} <font size=4><b>and</b></font> add to {{{-84}}}

Now replace the middle term {{{-84s}}} with {{{-42s-42s}}}. Remember, {{{-42}}} and {{{-42}}} add to {{{-84}}}. So this shows us that {{{-42s-42s=-84s}}}.

{{{49s^2+highlight(-42s-42s)+36}}} Replace the second term {{{-84s}}} with {{{-42s-42s}}}.

{{{(49s^2-42s)+(-42s+36)}}} Group the terms into two pairs.

{{{7s(7s-6)+(-42s+36)}}} Factor out the GCF {{{7s}}} from the first group.

{{{7s(7s-6)-6(7s-6)}}} Factor out {{{6}}} 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.

{{{(7s-6)(7s-6)}}} Combine like terms. Or factor out the common term {{{7s-6}}}

{{{(7s-6)^2}}} Condense the terms.

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

So {{{49s^2+36-84s}}} factors to {{{(7s-6)^2}}}.

In other words, {{{49s^2+36-84s=(7s-6)^2}}}.

Note: you can check the answer by expanding {{{(7s-6)^2}}} to get {{{49s^2+36-84s}}} or by graphing the original expression and the answer (the two graphs should be identical).

If you need more help, email me at <a href="mailto:jim_thompson5910@hotmail.com?Subject=Algebra%20Help">jim_thompson5910@hotmail.com</a>

Also, feel free to check out my <a href="http://www.freewebs.com/jimthompson5910/home.html">tutoring website</a>

Jim
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