Question 560570


Looking at the expression {{{x^2-18x+81}}}, we can see that the first coefficient is {{{1}}}, the second coefficient is {{{-18}}}, and the last term is {{{81}}}.



Now multiply the first coefficient {{{1}}} by the last term {{{81}}} to get {{{(1)(81)=81}}}.



Now the question is: what two whole numbers multiply to {{{81}}} (the previous product) <font size=4><b>and</b></font> add to the second coefficient {{{-18}}}?



To find these two numbers, we need to list <font size=4><b>all</b></font> of the factors of {{{81}}} (the previous product).



Factors of {{{81}}}:

1,3,9,27,81

-1,-3,-9,-27,-81



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



These factors pair up and multiply to {{{81}}}.

1*81 = 81
3*27 = 81
9*9 = 81
(-1)*(-81) = 81
(-3)*(-27) = 81
(-9)*(-9) = 81


Now let's add up each pair of factors to see if one pair adds to the middle coefficient {{{-18}}}:



<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>81</font></td><td  align="center"><font color=black>1+81=82</font></td></tr><tr><td  align="center"><font color=black>3</font></td><td  align="center"><font color=black>27</font></td><td  align="center"><font color=black>3+27=30</font></td></tr><tr><td  align="center"><font color=black>9</font></td><td  align="center"><font color=black>9</font></td><td  align="center"><font color=black>9+9=18</font></td></tr><tr><td  align="center"><font color=black>-1</font></td><td  align="center"><font color=black>-81</font></td><td  align="center"><font color=black>-1+(-81)=-82</font></td></tr><tr><td  align="center"><font color=black>-3</font></td><td  align="center"><font color=black>-27</font></td><td  align="center"><font color=black>-3+(-27)=-30</font></td></tr><tr><td  align="center"><font color=red>-9</font></td><td  align="center"><font color=red>-9</font></td><td  align="center"><font color=red>-9+(-9)=-18</font></td></tr></table>



From the table, we can see that the two numbers {{{-9}}} and {{{-9}}} add to {{{-18}}} (the middle coefficient).



So the two numbers {{{-9}}} and {{{-9}}} both multiply to {{{81}}} <font size=4><b>and</b></font> add to {{{-18}}}



Now replace the middle term {{{-18x}}} with {{{-9x-9x}}}. Remember, {{{-9}}} and {{{-9}}} add to {{{-18}}}. So this shows us that {{{-9x-9x=-18x}}}.



{{{x^2+highlight(-9x-9x)+81}}} Replace the second term {{{-18x}}} with {{{-9x-9x}}}.



{{{(x^2-9x)+(-9x+81)}}} Group the terms into two pairs.



{{{x(x-9)+(-9x+81)}}} Factor out the GCF {{{x}}} from the first group.



{{{x(x-9)-9(x-9)}}} Factor out {{{9}}} 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.



{{{(x-9)(x-9)}}} Combine like terms. Or factor out the common term {{{x-9}}}



{{{(x-9)^2}}} Condense the terms.



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Answer:



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



In other words, {{{x^2-18x+81=(x-9)^2}}}.



Note: you can check the answer by expanding {{{(x-9)^2}}} to get {{{x^2-18x+81}}} or by graphing the original expression and the answer (the two graphs should be identical).

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