Question 407015


Looking at {{{81a^2-36ab+4b^2}}} we can see that the first term is {{{81a^2}}} and the last term is {{{4b^2}}} where the coefficients are 81 and 4 respectively.


Now multiply the first coefficient 81 and the last coefficient 4 to get 324. Now what two numbers multiply to 324 and add to the  middle coefficient -36? Let's list all of the factors of 324:




Factors of 324:

1,2,3,4,6,9,12,18,27,36,54,81,108,162


-1,-2,-3,-4,-6,-9,-12,-18,-27,-36,-54,-81,-108,-162 ...List the negative factors as well. This will allow us to find all possible combinations


These factors pair up and multiply to 324

1*324

2*162

3*108

4*81

6*54

9*36

12*27

18*18

(-1)*(-324)

(-2)*(-162)

(-3)*(-108)

(-4)*(-81)

(-6)*(-54)

(-9)*(-36)

(-12)*(-27)

(-18)*(-18)


note: remember two negative numbers multiplied together make a positive number



Now which of these pairs add to -36? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -36


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">324</td><td>1+324=325</td></tr><tr><td align="center">2</td><td align="center">162</td><td>2+162=164</td></tr><tr><td align="center">3</td><td align="center">108</td><td>3+108=111</td></tr><tr><td align="center">4</td><td align="center">81</td><td>4+81=85</td></tr><tr><td align="center">6</td><td align="center">54</td><td>6+54=60</td></tr><tr><td align="center">9</td><td align="center">36</td><td>9+36=45</td></tr><tr><td align="center">12</td><td align="center">27</td><td>12+27=39</td></tr><tr><td align="center">18</td><td align="center">18</td><td>18+18=36</td></tr><tr><td align="center">-1</td><td align="center">-324</td><td>-1+(-324)=-325</td></tr><tr><td align="center">-2</td><td align="center">-162</td><td>-2+(-162)=-164</td></tr><tr><td align="center">-3</td><td align="center">-108</td><td>-3+(-108)=-111</td></tr><tr><td align="center">-4</td><td align="center">-81</td><td>-4+(-81)=-85</td></tr><tr><td align="center">-6</td><td align="center">-54</td><td>-6+(-54)=-60</td></tr><tr><td align="center">-9</td><td align="center">-36</td><td>-9+(-36)=-45</td></tr><tr><td align="center">-12</td><td align="center">-27</td><td>-12+(-27)=-39</td></tr><tr><td align="center">-18</td><td align="center">-18</td><td>-18+(-18)=-36</td></tr></table>



From this list we can see that -18 and -18 add up to -36 and multiply to 324



Now looking at the expression {{{81a^2-36ab+4b^2}}}, replace {{{-36ab}}} with {{{-18ab+-18ab}}} (notice {{{-18ab+-18ab}}} adds up to {{{-36ab}}}. So it is equivalent to {{{-36ab}}})


{{{81a^2+highlight(-18ab+-18ab)+4b^2}}}



Now let's factor {{{81a^2-18ab-18ab+4b^2}}} by grouping:



{{{(81a^2-18ab)+(-18ab+4b^2)}}} Group like terms



{{{9a(9a-2b)-2b(9a-2b)}}} Factor out the GCF of {{{9a}}} out of the first group. Factor out the GCF of {{{-2b}}} out of the second group



{{{(9a-2b)(9a-2b)}}} Since we have a common term of {{{9a-2b}}}, we can combine like terms


So {{{81a^2-18ab-18ab+4b^2}}} factors to {{{(9a-2b)(9a-2b)}}}



So this also means that {{{81a^2-36ab+4b^2}}} factors to {{{(9a-2b)(9a-2b)}}} (since {{{81a^2-36ab+4b^2}}} is equivalent to {{{81a^2-18ab-18ab+4b^2}}})



note:  {{{(9a-2b)(9a-2b)}}} is equivalent to  {{{(9a-2b)^2}}} since the term {{{9a-2b}}} occurs twice. So {{{81a^2-36ab+4b^2}}} also factors to {{{(9a-2b)^2}}}




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     Answer:

So {{{81a^2-36ab+4b^2}}} factors to {{{(9a-2b)^2}}}



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