x^2+4.1x-8.2=0

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Solution for x^2+4.1x-8.2=0 equation: 


Simplifying
x2 + 4.1x + -8.2 = 0

Reorder the terms:
-8.2 + 4.1x + x2 = 0

Solving
-8.2 + 4.1x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '8.2' to each side of the equation.
-8.2 + 4.1x + 8.2 + x2 = 0 + 8.2

Reorder the terms:
-8.2 + 8.2 + 4.1x + x2 = 0 + 8.2

Combine like terms: -8.2 + 8.2 = 0.0
0.0 + 4.1x + x2 = 0 + 8.2
4.1x + x2 = 0 + 8.2

Combine like terms: 0 + 8.2 = 8.2
4.1x + x2 = 8.2

The x term is 4.1x.  Take half its coefficient (2.05).
Square it (4.2025) and add it to both sides.

Add '4.2025' to each side of the equation.
4.1x + 4.2025 + x2 = 8.2 + 4.2025

Reorder the terms:
4.2025 + 4.1x + x2 = 8.2 + 4.2025

Combine like terms: 8.2 + 4.2025 = 12.4025
4.2025 + 4.1x + x2 = 12.4025

Factor a perfect square on the left side:
(x + 2.05)(x + 2.05) = 12.4025

Calculate the square root of the right side: 3.521718331

Break this problem into two subproblems by setting 
(x + 2.05) equal to 3.521718331 and -3.521718331.


Subproblem 1
x + 2.05 = 3.521718331

Simplifying
x + 2.05 = 3.521718331

Reorder the terms:
2.05 + x = 3.521718331

Solving
2.05 + x = 3.521718331

Solving for variable 'x'.

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

Add '-2.05' to each side of the equation.
2.05 + -2.05 + x = 3.521718331 + -2.05

Combine like terms: 2.05 + -2.05 = 0.00
0.00 + x = 3.521718331 + -2.05
x = 3.521718331 + -2.05

Combine like terms: 3.521718331 + -2.05 = 1.471718331
x = 1.471718331

Simplifying
x = 1.471718331


Subproblem 2
x + 2.05 = -3.521718331

Simplifying
x + 2.05 = -3.521718331

Reorder the terms:
2.05 + x = -3.521718331

Solving
2.05 + x = -3.521718331

Solving for variable 'x'.

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

Add '-2.05' to each side of the equation.
2.05 + -2.05 + x = -3.521718331 + -2.05

Combine like terms: 2.05 + -2.05 = 0.00
0.00 + x = -3.521718331 + -2.05
x = -3.521718331 + -2.05

Combine like terms: -3.521718331 + -2.05 = -5.571718331
x = -5.571718331

Simplifying
x = -5.571718331


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.471718331, -5.571718331}

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