0=x^2+10x+5

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Solution for 0=x^2+10x+5 equation: 


Simplifying
0 = x2 + 10x + 5

Reorder the terms:
0 = 5 + 10x + x2

Solving
0 = 5 + 10x + x2

Solving for variable 'x'.

Combine like terms: 0 + -5 = -5
-5 + -10x + -1x2 = 5 + 10x + x2 + -5 + -10x + -1x2

Reorder the terms:
-5 + -10x + -1x2 = 5 + -5 + 10x + -10x + x2 + -1x2

Combine like terms: 5 + -5 = 0
-5 + -10x + -1x2 = 0 + 10x + -10x + x2 + -1x2
-5 + -10x + -1x2 = 10x + -10x + x2 + -1x2

Combine like terms: 10x + -10x = 0
-5 + -10x + -1x2 = 0 + x2 + -1x2
-5 + -10x + -1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
-5 + -10x + -1x2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(5 + 10x + x2) = 0

Ignore the factor -1.

Subproblem 1
Set the factor '(5 + 10x + x2)' equal to zero and attempt to solve:

Simplifying
5 + 10x + x2 = 0

Solving
5 + 10x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-5' to each side of the equation.
5 + 10x + -5 + x2 = 0 + -5

Reorder the terms:
5 + -5 + 10x + x2 = 0 + -5

Combine like terms: 5 + -5 = 0
0 + 10x + x2 = 0 + -5
10x + x2 = 0 + -5

Combine like terms: 0 + -5 = -5
10x + x2 = -5

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 = -5 + 25

Reorder the terms:
25 + 10x + x2 = -5 + 25

Combine like terms: -5 + 25 = 20
25 + 10x + x2 = 20

Factor a perfect square on the left side:
(x + 5)(x + 5) = 20

Calculate the square root of the right side: 4.472135955

Break this problem into two subproblems by setting 
(x + 5) equal to 4.472135955 and -4.472135955.


Subproblem 1
x + 5 = 4.472135955

Simplifying
x + 5 = 4.472135955

Reorder the terms:
5 + x = 4.472135955

Solving
5 + x = 4.472135955

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 = 4.472135955 + -5

Combine like terms: 5 + -5 = 0
0 + x = 4.472135955 + -5
x = 4.472135955 + -5

Combine like terms: 4.472135955 + -5 = -0.527864045
x = -0.527864045

Simplifying
x = -0.527864045


Subproblem 2
x + 5 = -4.472135955

Simplifying
x + 5 = -4.472135955

Reorder the terms:
5 + x = -4.472135955

Solving
5 + x = -4.472135955

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 = -4.472135955 + -5

Combine like terms: 5 + -5 = 0
0 + x = -4.472135955 + -5
x = -4.472135955 + -5

Combine like terms: -4.472135955 + -5 = -9.472135955
x = -9.472135955

Simplifying
x = -9.472135955


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.527864045, -9.472135955}
Solution
x = {-0.527864045, -9.472135955}

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