Solution:

To find the value of the root of the equation 4x^2 - 2x - 1 = 0, we can use the quadratic formula. The quadratic formula states that for any quadratic equation in the form ax^2 + bx + c = 0, the roots can be found using the formula:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

In our equation, a = 4, b = -2, and c = -1. Plugging these values into the quadratic formula gives us:

x = (-(-2) ± sqrt((-2)^2 - 4(4)(-1))) / (2(4))
x = (2 ± sqrt(4 + 16)) / 8
x = (2 ± sqrt(20)) / 8
x = (2 ± 2sqrt(5)) / 8
x = (1 ± sqrt(5)) / 4

So the two roots of the equation are (1 + sqrt(5))/4 and (1 - sqrt(5))/4.

Summary:
- The quadratic formula can be used to find the roots of any quadratic equation in the form ax^2 + bx + c = 0.
- The formula is x = (-b ± sqrt(b^2 - 4ac)) / 2a.
- For the equation 4x^2 - 2x - 1 = 0, the roots are (1 + sqrt(5))/4 and (1 - sqrt(5))/4.
- The roots can be simplified further, if required.