2x^2+3-108+4x=20041

Solution for 2x^2+3-108+4x=20041 equation:

Simplifying
2x2 + 3 + -108 + 4x = 20041

Reorder the terms:
3 + -108 + 4x + 2x2 = 20041

Combine like terms: 3 + -108 = -105
-105 + 4x + 2x2 = 20041

Solving
-105 + 4x + 2x2 = 20041

Solving for variable 'x'.

Reorder the terms:
-105 + -20041 + 4x + 2x2 = 20041 + -20041

Combine like terms: -105 + -20041 = -20146
-20146 + 4x + 2x2 = 20041 + -20041

Combine like terms: 20041 + -20041 = 0
-20146 + 4x + 2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-10073 + 2x + x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-10073 + 2x + x2)' equal to zero and attempt to solve:

Simplifying
-10073 + 2x + x2 = 0

Solving
-10073 + 2x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '10073' to each side of the equation.
-10073 + 2x + 10073 + x2 = 0 + 10073

Reorder the terms:
-10073 + 10073 + 2x + x2 = 0 + 10073

Combine like terms: -10073 + 10073 = 0
0 + 2x + x2 = 0 + 10073
2x + x2 = 0 + 10073

Combine like terms: 0 + 10073 = 10073
2x + x2 = 10073

The x term is 2x.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2x + 1 + x2 = 10073 + 1

Reorder the terms:
1 + 2x + x2 = 10073 + 1

Combine like terms: 10073 + 1 = 10074
1 + 2x + x2 = 10074

Factor a perfect square on the left side:
(x + 1)(x + 1) = 10074

Calculate the square root of the right side: 100.369318021

Break this problem into two subproblems by setting 
(x + 1) equal to 100.369318021 and -100.369318021.

Subproblem 1
x + 1 = 100.369318021

Simplifying
x + 1 = 100.369318021

Reorder the terms:
1 + x = 100.369318021

Solving
1 + x = 100.369318021

Solving for variable 'x'.

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

Add '-1' to each side of the equation.
1 + -1 + x = 100.369318021 + -1

Combine like terms: 1 + -1 = 0
0 + x = 100.369318021 + -1
x = 100.369318021 + -1

Combine like terms: 100.369318021 + -1 = 99.369318021
x = 99.369318021

Simplifying
x = 99.369318021

Subproblem 2
x + 1 = -100.369318021

Simplifying
x + 1 = -100.369318021

Reorder the terms:
1 + x = -100.369318021

Solving
1 + x = -100.369318021

Solving for variable 'x'.

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

Add '-1' to each side of the equation.
1 + -1 + x = -100.369318021 + -1

Combine like terms: 1 + -1 = 0
0 + x = -100.369318021 + -1
x = -100.369318021 + -1

Combine like terms: -100.369318021 + -1 = -101.369318021
x = -101.369318021

Simplifying
x = -101.369318021

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {99.369318021, -101.369318021}
Solution
x = {99.369318021, -101.369318021}