The solution set of the equation x+ x=2 x , in the interval [0,2 ] is

Step-by-step solution

Equation: x+ x=2 x , in the interval [0,2 ] . Substitute x= 1 x and x= x x : gathered x x + 1 x =2 x x+1 x =2 x gathered Multiply through by x (assuming x 0 ): x+1=2 ^2 x Use ^2 x=1- ^2 x : gathered x+1=2 (1- ^2 x ) x+1=2-2 ^2 x gathered Rearrange into a quadratic in x : 2 ^2 x+ x-1=0 Solve the quadratic equation: gathered x= -1 1-4(2)(-1) 2(2) x= -1 9 4 x= -1+3 4 or x= -1-3 4 x= 1 2 or x=-1 gathered Find solutions in [0,2 ] : - For x= 1 2 , x= 6 , 5 6 . - For x=-1, x= 3 2 . Answer: 6 , 5 6 , 3 2 , Option 3.