(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 = -10x
(-7 + -10x + 8x2) = 1

Solving
-7 + -10x + 8x2 = 1

Solving for variable 'x'.

Reorder the terms:
-7 + -1 + -10x + 8x2 = 1 + -1

Combine like terms: -7 + -1 = -8
-8 + -10x + 8x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-8 + -10x + 8x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-4 + -5x + 4x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-4 + -5x + 4x2)' equal to zero and attempt to solve:

Simplifying
-4 + -5x + 4x2 = 0

Solving
-4 + -5x + 4x2 = 0

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

Divide each side by '4'.
-1 + -1.25x + x2 = 0

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + -1.25x + 1 + x2 = 0 + 1

Reorder the terms:
-1 + 1 + -1.25x + x2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -1.25x + x2 = 0 + 1
-1.25x + x2 = 0 + 1

Combine like terms: 0 + 1 = 1
-1.25x + x2 = 1

The x term is -1.25x.  Take half its coefficient (-0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
-1.25x + 0.390625 + x2 = 1 + 0.390625

Reorder the terms:
0.390625 + -1.25x + x2 = 1 + 0.390625

Combine like terms: 1 + 0.390625 = 1.390625
0.390625 + -1.25x + x2 = 1.390625

Factor a perfect square on the left side:
(x + -0.625)(x + -0.625) = 1.390625

Calculate the square root of the right side: 1.179247642

Break this problem into two subproblems by setting 
(x + -0.625) equal to 1.179247642 and -1.179247642.

Subproblem 1
x + -0.625 = 1.179247642

Simplifying
x + -0.625 = 1.179247642

Reorder the terms:
-0.625 + x = 1.179247642

Solving
-0.625 + x = 1.179247642

Solving for variable 'x'.

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

Add '0.625' to each side of the equation.
-0.625 + 0.625 + x = 1.179247642 + 0.625

Combine like terms: -0.625 + 0.625 = 0.000
0.000 + x = 1.179247642 + 0.625
x = 1.179247642 + 0.625

Combine like terms: 1.179247642 + 0.625 = 1.804247642
x = 1.804247642

Simplifying
x = 1.804247642

Subproblem 2
x + -0.625 = -1.179247642

Simplifying
x + -0.625 = -1.179247642

Reorder the terms:
-0.625 + x = -1.179247642

Solving
-0.625 + x = -1.179247642

Solving for variable 'x'.

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

Add '0.625' to each side of the equation.
-0.625 + 0.625 + x = -1.179247642 + 0.625

Combine like terms: -0.625 + 0.625 = 0.000
0.000 + x = -1.179247642 + 0.625
x = -1.179247642 + 0.625

Combine like terms: -1.179247642 + 0.625 = -0.554247642
x = -0.554247642

Simplifying
x = -0.554247642

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.804247642, -0.554247642}
Solution
x = {1.804247642, -0.554247642}