Find all real numbers x such that (3x - 27)^3 + (27x - 3)^3 = (3x + 27x - 30).
+128408 (3x - 27)^3 + (27x - 3)^3 = 30x - 30
Factor left side as a sum of cubes.....
[ (3x - 27 + 27x - 3) ] [ (3x - 27)^2 - (3x - 27)(27x - 3) + ( 27x - 3)^2 ] = 30x - 30
( 30x - 30) [ (3(x - 9))^2 - ( 3 (x - 9) * 3(9x - 1) ) + (3 (9x - 1))^2 ] = 30x -30
(30x - 30) [ 9 ( x^2 - 18x + 81) - ( 9(9x^2 - 82x + 9)) + (9 ( 81x^2 - 18x + 1)) = 30x - 30
9 (30x - 30) [ x^2 - 18x + 81 - 9x^2 + 82x - 9 + 81x^2 - 18x + 1 ] = 30x - 30
270 ( x - 1) [ 73x^2 + 46x + 73 ] = 30 (x - 1)
(270 - 30) ( x - 1) [ 73x^2 + 46x^2 + 73 ] = 0
(240) ( x - 1) [ 73x^2 + 46x + 73 ] = 0 divide through by 240
(x - 1) (73x^2 + 46x + 73 ] = 0
The discriminant of 73x^2 + 46x^2 + 73 = [ 46^2 - 4 (73)^2 ] = negative....so this factor has no real roots
The only real solution is
(x -1 ) = 0
x = 1
