Rewrite the equation using a single trigonometric function.

Step 1: Rewrite the equation using a single trigonometric function. The given equation is 2 ^2 x - x = 1. Using the identity ^2 x = 1 - ^2 x, substitute it into the equation: 2(1 - ^2 x) - x = 1 Step 2: Simplify and rearrange the equation into a quadratic form. 2 - 2^2 x - x = 1 0 = 2^2 x + x + 1 - 2 2^2 x + x - 1 = 0 Step 3: Solve the quadratic equation for x. Let y = x. The equation becomes 2y^2 + y - 1 = 0. Factor the quadratic equation: (2y - 1)(y + 1) = 0 This gives two possible values for y: 2y - 1 = 0 y = (1)/(2) y + 1 = 0 y = -1 So, x = (1)/(2) or x = -1. Step 4: Find the values of x for x = (1)/(2) in the interval 0 x 2. The principal value for x = (1)/(2) is x = ()/(6). Since x is positive, x can be in the first or second quadrant. In the first quadrant: x_1 = ()/(6). In the second quadrant: x_2 = - ()/(6) = (6 - )/(6) = (5)/(6). Step 5: Find the values of x for x = -1 in the interval 0 x 2. The value for which x = -1 is x_3 = (3)/(2). Step 6: List all solutions in the given interval. The values of x in the interval 0 x 2 are ()/(6), (5)/(6), and (3)/(2). The values of x are ()/(6), (5)/(6), (3)/(2). Send me the next one 📸