x^2+20x+83=0

Solution for x^2+20x+83=0 equation:

Simplifying
x2 + 20x + 83 = 0

Reorder the terms:
83 + 20x + x2 = 0

Solving
83 + 20x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '-83' to each side of the equation.
83 + 20x + -83 + x2 = 0 + -83

Reorder the terms:
83 + -83 + 20x + x2 = 0 + -83

Combine like terms: 83 + -83 = 0
0 + 20x + x2 = 0 + -83
20x + x2 = 0 + -83

Combine like terms: 0 + -83 = -83
20x + x2 = -83

The x term is 20x.  Take half its coefficient (10).
Square it (100) and add it to both sides.

Add '100' to each side of the equation.
20x + 100 + x2 = -83 + 100

Reorder the terms:
100 + 20x + x2 = -83 + 100

Combine like terms: -83 + 100 = 17
100 + 20x + x2 = 17

Factor a perfect square on the left side:
(x + 10)(x + 10) = 17

Calculate the square root of the right side: 4.123105626

Break this problem into two subproblems by setting 
(x + 10) equal to 4.123105626 and -4.123105626.

Subproblem 1
x + 10 = 4.123105626

Simplifying
x + 10 = 4.123105626

Reorder the terms:
10 + x = 4.123105626

Solving
10 + x = 4.123105626

Solving for variable 'x'.

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

Add '-10' to each side of the equation.
10 + -10 + x = 4.123105626 + -10

Combine like terms: 10 + -10 = 0
0 + x = 4.123105626 + -10
x = 4.123105626 + -10

Combine like terms: 4.123105626 + -10 = -5.876894374
x = -5.876894374

Simplifying
x = -5.876894374

Subproblem 2
x + 10 = -4.123105626

Simplifying
x + 10 = -4.123105626

Reorder the terms:
10 + x = -4.123105626

Solving
10 + x = -4.123105626

Solving for variable 'x'.

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

Add '-10' to each side of the equation.
10 + -10 + x = -4.123105626 + -10

Combine like terms: 10 + -10 = 0
0 + x = -4.123105626 + -10
x = -4.123105626 + -10

Combine like terms: -4.123105626 + -10 = -14.123105626
x = -14.123105626

Simplifying
x = -14.123105626

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-5.876894374, -14.123105626}