If 0 x 2 , then what is the number of values of x satisfying the equation x + x = 2 x ?

Step-by-step solution

The given equation is x + x = 2 x . Rewriting the equation in terms of x and x : x x + 1 x = 2 x For the equation to be defined, x 0 , which means x 2 . Thus, x [0, 2 ) . Multiplying both sides by x : x + 1 = 2 ^2 x Substituting ^2 x = 1 - ^2 x : x + 1 = 2(1 - ^2 x) 2 ^2 x + x - 1 = 0 Factoring the quadratic equation: (2 x - 1)( x + 1) = 0 This gives x = 1 2 or x = -1 . Since x [0, 2 ) , x 0 . Therefore, x = -1 is rejected. For x = 1 2 , the only solution in the given interval is x = 6 . Thus, there is exactly 1 va