The number of real solutions of the equation 3x x + 3x x = 4 - (x - )^2 is

Step-by-step solution

The left-hand side of the equation can be simplified using the triple-angle formulas 3x = 4 ^3 x - 3 x and 3x = 3 x - 4 ^3 x . 4 ^3 x - 3 x x + 3 x - 4 ^3 x x = (4 ^2 x - 3) + (3 - 4 ^2 x) This simplifies to: 4 ^2 x - 4 ^2 x = 4( ^2 x - ^2 x) = 4 2x This simplification requires the domain constraints x 0 and x 0 . The original equation becomes: 4 2x = 4 - (x - )^2 Analyze the maximum values of both sides. For the left-hand side, the maximum value of 4 2x is 4 . For the right-hand side, since (x - )^2 0 , the maximu