4x(x-1)=1

Solution for 4x(x-1)=1 equation:

Simplifying
4x(x + -1) = 1

Reorder the terms:
4x(-1 + x) = 1
(-1 * 4x + x * 4x) = 1
(-4x + 4x2) = 1

Solving
-4x + 4x2 = 1

Solving for variable 'x'.

Reorder the terms:
-1 + -4x + 4x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + -4x + 4x2 = 0

Begin completing the square.  Divide all terms by
4 the coefficient of the squared term: 

Divide each side by '4'.
-0.25 + -1x + x2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + -1x + 0.25 + x2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + -1x + x2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + -1x + x2 = 0 + 0.25
-1x + x2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
-1x + x2 = 0.25

The x term is -1x.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1x + 0.25 + x2 = 0.25 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 0.25 + 0.25

Combine like terms: 0.25 + 0.25 = 0.5
0.25 + -1x + x2 = 0.5

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 0.5

Calculate the square root of the right side: 0.707106781

Break this problem into two subproblems by setting 
(x + -0.5) equal to 0.707106781 and -0.707106781.

Subproblem 1
x + -0.5 = 0.707106781

Simplifying
x + -0.5 = 0.707106781

Reorder the terms:
-0.5 + x = 0.707106781

Solving
-0.5 + x = 0.707106781

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '0.5' to each side of the equation.
-0.5 + 0.5 + x = 0.707106781 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + x = 0.707106781 + 0.5
x = 0.707106781 + 0.5

Combine like terms: 0.707106781 + 0.5 = 1.207106781
x = 1.207106781

Simplifying
x = 1.207106781

Subproblem 2
x + -0.5 = -0.707106781

Simplifying
x + -0.5 = -0.707106781

Reorder the terms:
-0.5 + x = -0.707106781

Solving
-0.5 + x = -0.707106781

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '0.5' to each side of the equation.
-0.5 + 0.5 + x = -0.707106781 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + x = -0.707106781 + 0.5
x = -0.707106781 + 0.5

Combine like terms: -0.707106781 + 0.5 = -0.207106781
x = -0.207106781

Simplifying
x = -0.207106781

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.207106781, -0.207106781}