SOLUTION: Find all real numbers x such that (3x - 27)^3 + (27x - 3)^3 = (3x + 27x - 30)^3.

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Question 1209412: Find all real numbers x such that (3x - 27)^3 + (27x - 3)^3 = (3x + 27x - 30)^3.
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
 
%283x+-+27%29%5E3+%2B+%2827x+-+3%29%5E3%22%22=%22%22%283x+%2B+27x+-+30%29%5E3
%283%28x+-+9%29%29%5E3+%2B+%283%289x+-+1%29%29%5E3%22%22=%22%22%2830x+-+30%29%5E3
%283%28x+-+9%29%29%5E3+%2B+%283%289x+-+1%29%29%5E3%22%22=%22%2230%28x+-+1%29%5E3
27%28x+-+9%29%5E3+%2B+27%289x+-+1%29%5E3%22%22=%22%2227000%28x+-+1%29%5E3
%28x+-+9%29%5E3+%2B+%289x+-+1%29%5E3%22%22=%22%221000%28x+-+1%29%5E3

Factor the left side as the sum of two cubes:

%22%22=%22%221000%28x-1%29%5E3
%2810x-10%5E%22%22%29%28%28x-9%29%5E2-%28x-9%29%289x-1%29%2B%289x-1%29%5E2%29%22%22=%22%221000%28x-1%29%5E3
%22%22=%22%221000%28x-1%29%5E3
%2810%28x-1%29%5E%22%22%29%28x%5E2-18x%2B81-9x%5E2%2B82x-9%2B81x%5E2-18x%2B1%29%22%22=%22%221000%28x-1%5E%22%22%29%5E3
%2810%28x-1%29%5E%22%22%29%2873x%5E2%2B46x%2B73%29%22%22=%22%221000%28x-1%5E%22%22%29%5E3=0
%28x-1%5E%22%22%29%2873x%5E2%2B46x%2B73%29%22%22=%22%22100%28x-1%5E%22%22%29%5E3
%28x-1%5E%22%22%29%2873x%5E2%2B46x%2B73%29-100%28x-1%5E%22%22%29%5E3%22%22=%22%220
%28x-1%5E%22%22%29%28%2873x%5E2%2B46x%2B73%29%5E%22%22-100%28x-1%5E%22%22%29%5E2%29%22%22=%22%220
%28x-1%5E%22%22%29%2873x%5E2%2B46x%2B73-100%28x%5E2-2x%2B1%29%29%22%22=%22%220
%28x-1%5E%22%22%29%2873x%5E2%2B46x%2B73-100x%5E2%2B200x-100%29%22%22=%22%220
%28x-1%5E%22%22%29%28-27x%5E2%2B246x-27%29%22%22=%22%220
Divide both sides by -3, since the 2nd factor on the left
is divisible by -3
%28x-1%5E%22%22%29%289x%5E2-82x%2B9%29%22%22=%22%220
%28x-1%29%289x-1%29%28x-9%29%22%22=%22%220
x-1=0; 9x-1=0  ; x-9=0
  x=1;   9x=1  ;   x=9
          x=1%2F9;

So the three solutions are 1, 1%2F9, and 9.

Edwin