(x-3)(2x+1)=1

Solution for (x-3)(2x+1)=1 equation:

Simplifying
(x + -3)(2x + 1) = 1

Reorder the terms:
(-3 + x)(2x + 1) = 1

Reorder the terms:
(-3 + x)(1 + 2x) = 1

Multiply (-3 + x) * (1 + 2x)
(-3(1 + 2x) + x(1 + 2x)) = 1
((1 * -3 + 2x * -3) + x(1 + 2x)) = 1
((-3 + -6x) + x(1 + 2x)) = 1
(-3 + -6x + (1 * x + 2x * x)) = 1
(-3 + -6x + (1x + 2x2)) = 1

Combine like terms: -6x + 1x = -5x
(-3 + -5x + 2x2) = 1

Solving
-3 + -5x + 2x2 = 1

Solving for variable 'x'.

Reorder the terms:
-3 + -1 + -5x + 2x2 = 1 + -1

Combine like terms: -3 + -1 = -4
-4 + -5x + 2x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-4 + -5x + 2x2 = 0

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

Divide each side by '2'.
-2 + -2.5x + x2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -2.5x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + -2.5x + x2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -2.5x + x2 = 0 + 2
-2.5x + x2 = 0 + 2

Combine like terms: 0 + 2 = 2
-2.5x + x2 = 2

The x term is -2.5x.  Take half its coefficient (-1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
-2.5x + 1.5625 + x2 = 2 + 1.5625

Reorder the terms:
1.5625 + -2.5x + x2 = 2 + 1.5625

Combine like terms: 2 + 1.5625 = 3.5625
1.5625 + -2.5x + x2 = 3.5625

Factor a perfect square on the left side:
(x + -1.25)(x + -1.25) = 3.5625

Calculate the square root of the right side: 1.887458609

Break this problem into two subproblems by setting 
(x + -1.25) equal to 1.887458609 and -1.887458609.

Subproblem 1
x + -1.25 = 1.887458609

Simplifying
x + -1.25 = 1.887458609

Reorder the terms:
-1.25 + x = 1.887458609

Solving
-1.25 + x = 1.887458609

Solving for variable 'x'.

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

Add '1.25' to each side of the equation.
-1.25 + 1.25 + x = 1.887458609 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + x = 1.887458609 + 1.25
x = 1.887458609 + 1.25

Combine like terms: 1.887458609 + 1.25 = 3.137458609
x = 3.137458609

Simplifying
x = 3.137458609

Subproblem 2
x + -1.25 = -1.887458609

Simplifying
x + -1.25 = -1.887458609

Reorder the terms:
-1.25 + x = -1.887458609

Solving
-1.25 + x = -1.887458609

Solving for variable 'x'.

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

Add '1.25' to each side of the equation.
-1.25 + 1.25 + x = -1.887458609 + 1.25

Combine like terms: -1.25 + 1.25 = 0.00
0.00 + x = -1.887458609 + 1.25
x = -1.887458609 + 1.25

Combine like terms: -1.887458609 + 1.25 = -0.637458609
x = -0.637458609

Simplifying
x = -0.637458609

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.137458609, -0.637458609}