The sum of all distinct solutions of the equation 3x + x = 2x in the interval x [0, 3 ] is

Step-by-step solution

Given the equation: 3x + x = 2x Using the sum-to-product formula C + D = 2 ( C+D 2 ) ( C-D 2 ) on the left hand side, and the double angle formula 2x = 2 x x on the right hand side, we get: 2 2x x = 2 x x Bringing all terms to one side and factoring: 2 x ( 2x - x) = 0 This gives two branches of solutions: Case 1: x = 0 In the interval [0, 3 ] , the solutions are x = 2 , 3 2 , 5 2 . Case 2: 2x - x = 0 Using 2x = 1 - 2 ^2 x , we have: 1 - 2 ^2 x - x = 0 2 ^2 x + x - 1 = 0 (2 x - 1)( x + 1) = 0 This yields x = 1 2 or