To solve this problem, we can use the Newton-Raphson method, which is an iterative method for finding successively better approximations to the roots (or zeroes) of a real-valued function.
The Newton-Raphson method is based on the idea that if we have an initial guess x₁ for the root of the equation f(x) = 0, then a better approximation x₂ can be found using the formula:
x₂ = x₁ - f(x₁) / f'(x₁)
Where f'(x) is the derivative of the function f(x) with respect to x.
In this case, the equation we want to solve is -4x + cos(x) + 2 = 0. To apply the Newton-Raphson method, we need to find the derivative of the function f(x) = -4x + cos(x) + 2.
The derivative of f(x) is f'(x) = -4 - sin(x).
Now, we can use the given initial guess x₁=0.5 and the formula for the Newton-Raphson method to find the better approximation x₂.
x₂ = x₁ - (-4x₁ + cos(x₁) + 2) / (-4 - sin(x₁))
After calculating x₂, we can use it as the new initial guess to find x₃, and continue this process until we reach the desired level of accuracy.
By following these steps, we can determine the real root of the equation -4x + cos(x) + 2 = 0 using the Newton-Raphson method up to four decimal places.