(x+4)(x+5)=1

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Solution for (x+4)(x+5)=1 equation: 


Simplifying
(x + 4)(x + 5) = 1

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

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

Multiply (4 + x) * (5 + x)
(4(5 + x) + x(5 + x)) = 1
((5 * 4 + x * 4) + x(5 + x)) = 1
((20 + 4x) + x(5 + x)) = 1
(20 + 4x + (5 * x + x * x)) = 1
(20 + 4x + (5x + x2)) = 1

Combine like terms: 4x + 5x = 9x
(20 + 9x + x2) = 1

Solving
20 + 9x + x2 = 1

Solving for variable 'x'.

Reorder the terms:
20 + -1 + 9x + x2 = 1 + -1

Combine like terms: 20 + -1 = 19
19 + 9x + x2 = 1 + -1

Combine like terms: 1 + -1 = 0
19 + 9x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-19' to each side of the equation.
19 + 9x + -19 + x2 = 0 + -19

Reorder the terms:
19 + -19 + 9x + x2 = 0 + -19

Combine like terms: 19 + -19 = 0
0 + 9x + x2 = 0 + -19
9x + x2 = 0 + -19

Combine like terms: 0 + -19 = -19
9x + x2 = -19

The x term is 9x.  Take half its coefficient (4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
9x + 20.25 + x2 = -19 + 20.25

Reorder the terms:
20.25 + 9x + x2 = -19 + 20.25

Combine like terms: -19 + 20.25 = 1.25
20.25 + 9x + x2 = 1.25

Factor a perfect square on the left side:
(x + 4.5)(x + 4.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(x + 4.5) equal to 1.118033989 and -1.118033989.


Subproblem 1
x + 4.5 = 1.118033989

Simplifying
x + 4.5 = 1.118033989

Reorder the terms:
4.5 + x = 1.118033989

Solving
4.5 + x = 1.118033989

Solving for variable 'x'.

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

Add '-4.5' to each side of the equation.
4.5 + -4.5 + x = 1.118033989 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + x = 1.118033989 + -4.5
x = 1.118033989 + -4.5

Combine like terms: 1.118033989 + -4.5 = -3.381966011
x = -3.381966011

Simplifying
x = -3.381966011


Subproblem 2
x + 4.5 = -1.118033989

Simplifying
x + 4.5 = -1.118033989

Reorder the terms:
4.5 + x = -1.118033989

Solving
4.5 + x = -1.118033989

Solving for variable 'x'.

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

Add '-4.5' to each side of the equation.
4.5 + -4.5 + x = -1.118033989 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + x = -1.118033989 + -4.5
x = -1.118033989 + -4.5

Combine like terms: -1.118033989 + -4.5 = -5.618033989
x = -5.618033989

Simplifying
x = -5.618033989


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-3.381966011, -5.618033989}

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