(2x+1)(x+5)=1

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Solution for (2x+1)(x+5)=1 equation: 


Simplifying
(2x + 1)(x + 5) = 1

Reorder the terms:
(1 + 2x)(x + 5) = 1

Reorder the terms:
(1 + 2x)(5 + x) = 1

Multiply (1 + 2x) * (5 + x)
(1(5 + x) + 2x * (5 + x)) = 1
((5 * 1 + x * 1) + 2x * (5 + x)) = 1
((5 + 1x) + 2x * (5 + x)) = 1
(5 + 1x + (5 * 2x + x * 2x)) = 1
(5 + 1x + (10x + 2x2)) = 1

Combine like terms: 1x + 10x = 11x
(5 + 11x + 2x2) = 1

Solving
5 + 11x + 2x2 = 1

Solving for variable 'x'.

Reorder the terms:
5 + -1 + 11x + 2x2 = 1 + -1

Combine like terms: 5 + -1 = 4
4 + 11x + 2x2 = 1 + -1

Combine like terms: 1 + -1 = 0
4 + 11x + 2x2 = 0

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

Divide each side by '2'.
2 + 5.5x + x2 = 0

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + 5.5x + -2 + x2 = 0 + -2

Reorder the terms:
2 + -2 + 5.5x + x2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + 5.5x + x2 = 0 + -2
5.5x + x2 = 0 + -2

Combine like terms: 0 + -2 = -2
5.5x + x2 = -2

The x term is 5.5x.  Take half its coefficient (2.75).
Square it (7.5625) and add it to both sides.

Add '7.5625' to each side of the equation.
5.5x + 7.5625 + x2 = -2 + 7.5625

Reorder the terms:
7.5625 + 5.5x + x2 = -2 + 7.5625

Combine like terms: -2 + 7.5625 = 5.5625
7.5625 + 5.5x + x2 = 5.5625

Factor a perfect square on the left side:
(x + 2.75)(x + 2.75) = 5.5625

Calculate the square root of the right side: 2.358495283

Break this problem into two subproblems by setting 
(x + 2.75) equal to 2.358495283 and -2.358495283.


Subproblem 1
x + 2.75 = 2.358495283

Simplifying
x + 2.75 = 2.358495283

Reorder the terms:
2.75 + x = 2.358495283

Solving
2.75 + x = 2.358495283

Solving for variable 'x'.

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

Add '-2.75' to each side of the equation.
2.75 + -2.75 + x = 2.358495283 + -2.75

Combine like terms: 2.75 + -2.75 = 0.00
0.00 + x = 2.358495283 + -2.75
x = 2.358495283 + -2.75

Combine like terms: 2.358495283 + -2.75 = -0.391504717
x = -0.391504717

Simplifying
x = -0.391504717


Subproblem 2
x + 2.75 = -2.358495283

Simplifying
x + 2.75 = -2.358495283

Reorder the terms:
2.75 + x = -2.358495283

Solving
2.75 + x = -2.358495283

Solving for variable 'x'.

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

Add '-2.75' to each side of the equation.
2.75 + -2.75 + x = -2.358495283 + -2.75

Combine like terms: 2.75 + -2.75 = 0.00
0.00 + x = -2.358495283 + -2.75
x = -2.358495283 + -2.75

Combine like terms: -2.358495283 + -2.75 = -5.108495283
x = -5.108495283

Simplifying
x = -5.108495283


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.391504717, -5.108495283}

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