Question 133168
I'll do the first two to get you started



# 1


{{{9-x^2}}} Start with the given expression


{{{(3)^2-x^2}}} Rewrite {{{9}}} as {{{(3)^2}}}


{{{(3)^2-(x)^2}}} Rewrite {{{x^2}}} as {{{(x)^2}}}



Now use the difference of squares. Remember, the difference of squares formula is {{{A^2-B^2=(A+B)(A-B)}}} where in this case {{{A=3}}} and {{{B=x}}}


{{{9-x^2=(3+x)(3-x)}}} Plug in {{{A=3}}} and {{{B=x}}}




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Answer:


So the expression


{{{9-x^2}}}


factors to


{{{(3+x)(3-x)}}}


Notice that if you foil the factored expression, you get the original expression. This verifies our answer.



<hr>


{{{4x^2+81-36x}}} Start with the given expression



{{{4x^2-36x+81}}} Rearrange the terms





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


Now multiply the first coefficient 4 and the last coefficient 81 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 {{{4x^2-36x+81}}}, replace {{{-36x}}} with {{{-18x+-18x}}} (notice {{{-18x+-18x}}} adds up to {{{-36x}}}. So it is equivalent to {{{-36x}}})


{{{4x^2+highlight(-18x+-18x)+81}}}



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



{{{(4x^2-18x)+(-18x+81)}}} Group like terms



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



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


So {{{4x^2-18x-18x+81}}} factors to {{{(2x-9)(2x-9)}}}



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



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




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     Answer:

So {{{4x^2-36x+81}}} factors to {{{(2x-9)^2}}}