10.0x^2+20.1x+5.1=0

Solution for 10.0x^2+20.1x+5.1=0 equation:

Simplifying
10.0x2 + 20.1x + 5.1 = 0

Reorder the terms:
5.1 + 20.1x + 10.0x2 = 0

Solving
5.1 + 20.1x + 10.0x2 = 0

Solving for variable 'x'.

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

Divide each side by '10.0'.
0.51 + 2.01x + x2 = 0

Move the constant term to the right:

Add '-0.51' to each side of the equation.
0.51 + 2.01x + -0.51 + x2 = 0 + -0.51

Reorder the terms:
0.51 + -0.51 + 2.01x + x2 = 0 + -0.51

Combine like terms: 0.51 + -0.51 = 0.00
0.00 + 2.01x + x2 = 0 + -0.51
2.01x + x2 = 0 + -0.51

Combine like terms: 0 + -0.51 = -0.51
2.01x + x2 = -0.51

The x term is 2.01x.  Take half its coefficient (1.005).
Square it (1.010025) and add it to both sides.

Add '1.010025' to each side of the equation.
2.01x + 1.010025 + x2 = -0.51 + 1.010025

Reorder the terms:
1.010025 + 2.01x + x2 = -0.51 + 1.010025

Combine like terms: -0.51 + 1.010025 = 0.500025
1.010025 + 2.01x + x2 = 0.500025

Factor a perfect square on the left side:
(x + 1.005)(x + 1.005) = 0.500025

Calculate the square root of the right side: 0.707124459

Break this problem into two subproblems by setting 
(x + 1.005) equal to 0.707124459 and -0.707124459.

Subproblem 1
x + 1.005 = 0.707124459

Simplifying
x + 1.005 = 0.707124459

Reorder the terms:
1.005 + x = 0.707124459

Solving
1.005 + x = 0.707124459

Solving for variable 'x'.

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

Add '-1.005' to each side of the equation.
1.005 + -1.005 + x = 0.707124459 + -1.005

Combine like terms: 1.005 + -1.005 = 0.000
0.000 + x = 0.707124459 + -1.005
x = 0.707124459 + -1.005

Combine like terms: 0.707124459 + -1.005 = -0.297875541
x = -0.297875541

Simplifying
x = -0.297875541

Subproblem 2
x + 1.005 = -0.707124459

Simplifying
x + 1.005 = -0.707124459

Reorder the terms:
1.005 + x = -0.707124459

Solving
1.005 + x = -0.707124459

Solving for variable 'x'.

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

Add '-1.005' to each side of the equation.
1.005 + -1.005 + x = -0.707124459 + -1.005

Combine like terms: 1.005 + -1.005 = 0.000
0.000 + x = -0.707124459 + -1.005
x = -0.707124459 + -1.005

Combine like terms: -0.707124459 + -1.005 = -1.712124459
x = -1.712124459

Simplifying
x = -1.712124459

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.297875541, -1.712124459}