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

Step-by-step solution

The given equation is x^2 - 4x + 3 = ( x 2 ) . Let f(x) = x^2 - 4x + 3 = (x-1)(x-3) and g(x) = ( x 2 ) . The graph of f(x) is an upward-opening parabola with roots at x = 1 and x = 3 , and its vertex at (2, -1) . The graph of g(x) is a sine wave with period 4 and amplitude 1 . Let us analyze the number of intersections by dividing the domain into sub-intervals: 1. For x 0 : f(x) 3 and g(x) 1 . No intersection. 2. In (0, 1) : f(x) strictly decreases from 3 to 0 , while g(x) strictly increases from 0 to 1 . Since f(0