(2x-1)(4x-7)=1

Solution for (2x-1)(4x-7)=1 equation:

Simplifying
(2x + -1)(4x + -7) = 1

Reorder the terms:
(-1 + 2x)(4x + -7) = 1

Reorder the terms:
(-1 + 2x)(-7 + 4x) = 1

Multiply (-1 + 2x) * (-7 + 4x)
(-1(-7 + 4x) + 2x * (-7 + 4x)) = 1
((-7 * -1 + 4x * -1) + 2x * (-7 + 4x)) = 1
((7 + -4x) + 2x * (-7 + 4x)) = 1
(7 + -4x + (-7 * 2x + 4x * 2x)) = 1
(7 + -4x + (-14x + 8x2)) = 1

Combine like terms: -4x + -14x = -18x
(7 + -18x + 8x2) = 1

Solving
7 + -18x + 8x2 = 1

Solving for variable 'x'.

Reorder the terms:
7 + -1 + -18x + 8x2 = 1 + -1

Combine like terms: 7 + -1 = 6
6 + -18x + 8x2 = 1 + -1

Combine like terms: 1 + -1 = 0
6 + -18x + 8x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(3 + -9x + 4x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(3 + -9x + 4x2)' equal to zero and attempt to solve:

Simplifying
3 + -9x + 4x2 = 0

Solving
3 + -9x + 4x2 = 0

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

Divide each side by '4'.
0.75 + -2.25x + x2 = 0

Move the constant term to the right:

Add '-0.75' to each side of the equation.
0.75 + -2.25x + -0.75 + x2 = 0 + -0.75

Reorder the terms:
0.75 + -0.75 + -2.25x + x2 = 0 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + -2.25x + x2 = 0 + -0.75
-2.25x + x2 = 0 + -0.75

Combine like terms: 0 + -0.75 = -0.75
-2.25x + x2 = -0.75

The x term is -2.25x.  Take half its coefficient (-1.125).
Square it (1.265625) and add it to both sides.

Add '1.265625' to each side of the equation.
-2.25x + 1.265625 + x2 = -0.75 + 1.265625

Reorder the terms:
1.265625 + -2.25x + x2 = -0.75 + 1.265625

Combine like terms: -0.75 + 1.265625 = 0.515625
1.265625 + -2.25x + x2 = 0.515625

Factor a perfect square on the left side:
(x + -1.125)(x + -1.125) = 0.515625

Calculate the square root of the right side: 0.718070331

Break this problem into two subproblems by setting 
(x + -1.125) equal to 0.718070331 and -0.718070331.

Subproblem 1
x + -1.125 = 0.718070331

Simplifying
x + -1.125 = 0.718070331

Reorder the terms:
-1.125 + x = 0.718070331

Solving
-1.125 + x = 0.718070331

Solving for variable 'x'.

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

Add '1.125' to each side of the equation.
-1.125 + 1.125 + x = 0.718070331 + 1.125

Combine like terms: -1.125 + 1.125 = 0.000
0.000 + x = 0.718070331 + 1.125
x = 0.718070331 + 1.125

Combine like terms: 0.718070331 + 1.125 = 1.843070331
x = 1.843070331

Simplifying
x = 1.843070331

Subproblem 2
x + -1.125 = -0.718070331

Simplifying
x + -1.125 = -0.718070331

Reorder the terms:
-1.125 + x = -0.718070331

Solving
-1.125 + x = -0.718070331

Solving for variable 'x'.

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

Add '1.125' to each side of the equation.
-1.125 + 1.125 + x = -0.718070331 + 1.125

Combine like terms: -1.125 + 1.125 = 0.000
0.000 + x = -0.718070331 + 1.125
x = -0.718070331 + 1.125

Combine like terms: -0.718070331 + 1.125 = 0.406929669
x = 0.406929669

Simplifying
x = 0.406929669

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.843070331, 0.406929669}
Solution
x = {1.843070331, 0.406929669}