We have to find the values of x that satisfy: 4x^2 -4x -1 = 0

For a quadratic equation the roots can be expressed as

x1 = [-b + sqrt (b^2 - 4ac)]/2a

=> [4 + sqrt ( 16 + 16)]/8

=> 1/2 + sqrt 2 / 2

x2 = x1 = [-b + sqrt (b^2 - 4ac)]/2a

=> 1/2 - sqrt 2 /2

The values of x which satisfy 4x^2 -4x -1 = 0 are 1/2 + sqrt 2 / 2 and 1/2 - sqrt 2 / 2.

Given the quadratic equation 4x^2 -4x -1 =0

We need to find the solution or the roots of the equation.

We will use the quadratic formula.

==> a = 4     b= -4    c = -1

==> x = (-b + - sqrt(b^2 -4ac)/ 2a

==> x1 = (4 + sqrt(16-4*4*-1) / 2*4

            = (4+sqrt(32) / 8

            = (4+ 4sqrt2)/8 = (1/2) + sqrt2 /2

==> x1= (1/2) + sqrt2 / 2

==> x2 = (1/2) - sqrt2 / 2