If the solution of the equation _ x ( x) + 4 _ x ( x) = 1, x (0, 2 ) 4 is x = ⁻¹ ( + 2 ) , where , are integers, then + is equal to:

Step-by-step solution

Given equation: _ x ( x) + 4 _ x ( x) = 1 We know that x = x x and x = x x . Substituting these into the equation: _ x ( x x ) + 4 _ x ( x x ) = 1 Using the quotient rule for logarithms: (1 - _ x ( x) ) + 4 (1 - _ x ( x) ) = 1 Let t = _ x ( x) . Then _ x ( x) = 1 t . The equation becomes: 1 - t + 4 - 4 t = 1 4 - t - 4 t = 0 t^2 - 4t + 4 = 0 (t - 2)^2 = 0 t = 2 So, _ x ( x) = 2 x = ^2 x Using the identity ^2 x = ^2 x - 1 : x = ^2 x - 1 ^2 x - x - 1 = 0 Solving this quadratic equation for x : x = 1 1 - 4(1)(-1) 2 = 1