x(-2x-13)=1

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Solution for x(-2x-13)=1 equation: 


Simplifying
x(-2x + -13) = 1

Reorder the terms:
x(-13 + -2x) = 1
(-13 * x + -2x * x) = 1
(-13x + -2x2) = 1

Solving
-13x + -2x2 = 1

Solving for variable 'x'.

Reorder the terms:
-1 + -13x + -2x2 = 1 + -1

Combine like terms: 1 + -1 = 0
-1 + -13x + -2x2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(1 + 13x + 2x2) = 0

Ignore the factor -1.

Subproblem 1
Set the factor '(1 + 13x + 2x2)' equal to zero and attempt to solve:

Simplifying
1 + 13x + 2x2 = 0

Solving
1 + 13x + 2x2 = 0

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

Divide each side by '2'.
0.5 + 6.5x + x2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + 6.5x + -0.5 + x2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + 6.5x + x2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + 6.5x + x2 = 0 + -0.5
6.5x + x2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
6.5x + x2 = -0.5

The x term is 6.5x.  Take half its coefficient (3.25).
Square it (10.5625) and add it to both sides.

Add '10.5625' to each side of the equation.
6.5x + 10.5625 + x2 = -0.5 + 10.5625

Reorder the terms:
10.5625 + 6.5x + x2 = -0.5 + 10.5625

Combine like terms: -0.5 + 10.5625 = 10.0625
10.5625 + 6.5x + x2 = 10.0625

Factor a perfect square on the left side:
(x + 3.25)(x + 3.25) = 10.0625

Calculate the square root of the right side: 3.172144385

Break this problem into two subproblems by setting 
(x + 3.25) equal to 3.172144385 and -3.172144385.


Subproblem 1
x + 3.25 = 3.172144385

Simplifying
x + 3.25 = 3.172144385

Reorder the terms:
3.25 + x = 3.172144385

Solving
3.25 + x = 3.172144385

Solving for variable 'x'.

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

Add '-3.25' to each side of the equation.
3.25 + -3.25 + x = 3.172144385 + -3.25

Combine like terms: 3.25 + -3.25 = 0.00
0.00 + x = 3.172144385 + -3.25
x = 3.172144385 + -3.25

Combine like terms: 3.172144385 + -3.25 = -0.077855615
x = -0.077855615

Simplifying
x = -0.077855615


Subproblem 2
x + 3.25 = -3.172144385

Simplifying
x + 3.25 = -3.172144385

Reorder the terms:
3.25 + x = -3.172144385

Solving
3.25 + x = -3.172144385

Solving for variable 'x'.

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

Add '-3.25' to each side of the equation.
3.25 + -3.25 + x = -3.172144385 + -3.25

Combine like terms: 3.25 + -3.25 = 0.00
0.00 + x = -3.172144385 + -3.25
x = -3.172144385 + -3.25

Combine like terms: -3.172144385 + -3.25 = -6.422144385
x = -6.422144385

Simplifying
x = -6.422144385


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.077855615, -6.422144385}
Solution
x = {-0.077855615, -6.422144385}

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