2x^2+x=205

Solution for 2x^2+x=205 equation:

Simplifying
2x2 + x = 205

Reorder the terms:
x + 2x2 = 205

Solving
x + 2x2 = 205

Solving for variable 'x'.

Reorder the terms:
-205 + x + 2x2 = 205 + -205

Combine like terms: 205 + -205 = 0
-205 + x + 2x2 = 0

Begin completing the square.  Divide all terms by
2 the coefficient of the squared term: 

Divide each side by '2'.
-102.5 + 0.5x + x2 = 0

Move the constant term to the right:

Add '102.5' to each side of the equation.
-102.5 + 0.5x + 102.5 + x2 = 0 + 102.5

Reorder the terms:
-102.5 + 102.5 + 0.5x + x2 = 0 + 102.5

Combine like terms: -102.5 + 102.5 = 0.0
0.0 + 0.5x + x2 = 0 + 102.5
0.5x + x2 = 0 + 102.5

Combine like terms: 0 + 102.5 = 102.5
0.5x + x2 = 102.5

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.5x + 0.25 + x2 = 102.5 + 0.25

Reorder the terms:
0.25 + 0.5x + x2 = 102.5 + 0.25

Combine like terms: 102.5 + 0.25 = 102.75
0.25 + 0.5x + x2 = 102.75

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 102.75

Calculate the square root of the right side: 10.136567466

Break this problem into two subproblems by setting 
(x + 0.5) equal to 10.136567466 and -10.136567466.

Subproblem 1
x + 0.5 = 10.136567466

Simplifying
x + 0.5 = 10.136567466

Reorder the terms:
0.5 + x = 10.136567466

Solving
0.5 + x = 10.136567466

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 10.136567466 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 10.136567466 + -0.5
x = 10.136567466 + -0.5

Combine like terms: 10.136567466 + -0.5 = 9.636567466
x = 9.636567466

Simplifying
x = 9.636567466

Subproblem 2
x + 0.5 = -10.136567466

Simplifying
x + 0.5 = -10.136567466

Reorder the terms:
0.5 + x = -10.136567466

Solving
0.5 + x = -10.136567466

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -10.136567466 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -10.136567466 + -0.5
x = -10.136567466 + -0.5

Combine like terms: -10.136567466 + -0.5 = -10.636567466
x = -10.636567466

Simplifying
x = -10.636567466

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {9.636567466, -10.636567466}