-x^2-10x-23=0

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Solution for -x^2-10x-23=0 equation: 


Simplifying
-1x2 + -10x + -23 = 0

Reorder the terms:
-23 + -10x + -1x2 = 0

Solving
-23 + -10x + -1x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '-1'.
-1(23 + 10x + x2) = 0

Ignore the factor -1.

Subproblem 1
Set the factor '(23 + 10x + x2)' equal to zero and attempt to solve:

Simplifying
23 + 10x + x2 = 0

Solving
23 + 10x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-23' to each side of the equation.
23 + 10x + -23 + x2 = 0 + -23

Reorder the terms:
23 + -23 + 10x + x2 = 0 + -23

Combine like terms: 23 + -23 = 0
0 + 10x + x2 = 0 + -23
10x + x2 = 0 + -23

Combine like terms: 0 + -23 = -23
10x + x2 = -23

The x term is 10x.  Take half its coefficient (5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
10x + 25 + x2 = -23 + 25

Reorder the terms:
25 + 10x + x2 = -23 + 25

Combine like terms: -23 + 25 = 2
25 + 10x + x2 = 2

Factor a perfect square on the left side:
(x + 5)(x + 5) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(x + 5) equal to 1.414213562 and -1.414213562.


Subproblem 1
x + 5 = 1.414213562

Simplifying
x + 5 = 1.414213562

Reorder the terms:
5 + x = 1.414213562

Solving
5 + x = 1.414213562

Solving for variable 'x'.

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

Add '-5' to each side of the equation.
5 + -5 + x = 1.414213562 + -5

Combine like terms: 5 + -5 = 0
0 + x = 1.414213562 + -5
x = 1.414213562 + -5

Combine like terms: 1.414213562 + -5 = -3.585786438
x = -3.585786438

Simplifying
x = -3.585786438


Subproblem 2
x + 5 = -1.414213562

Simplifying
x + 5 = -1.414213562

Reorder the terms:
5 + x = -1.414213562

Solving
5 + x = -1.414213562

Solving for variable 'x'.

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

Add '-5' to each side of the equation.
5 + -5 + x = -1.414213562 + -5

Combine like terms: 5 + -5 = 0
0 + x = -1.414213562 + -5
x = -1.414213562 + -5

Combine like terms: -1.414213562 + -5 = -6.414213562
x = -6.414213562

Simplifying
x = -6.414213562


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-3.585786438, -6.414213562}
Solution
x = {-3.585786438, -6.414213562}

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