The sum of all valid real solutions x of the equation ^3 ( x 2 ) - ( 3 x 2 ) ( x 2 ) + ^3 ( x 2 ) + ( 3 x 2 ) ( x 2 ) = x^3 - 6x^2 + 7x + 5 is

Step-by-step solution

Let A = x 2 . The left-hand side of the equation can be simplified using the triple-angle formulas 3A = 4 ^3 A - 3 A and 3A = 3 A - 4 ^3 A . The first term becomes: ^3 A - (4 ^3 A - 3 A) A = 3 A - 3 ^3 A A = 3 - 3 ^2 A The second term becomes: ^3 A + (3 A - 4 ^3 A) A = 3 A - 3 ^3 A A = 3 - 3 ^2 A Adding these two terms gives: (3 - 3 ^2 A) + (3 - 3 ^2 A) = 6 - 3( ^2 A + ^2 A) = 6 - 3(1) = 3 This simplification is valid provided A 0 and A 0 , which means A n 2 for any integer n . Since A = x 2 , this requires x to no