18.85=0.1*x+4*x^2

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Solution for 18.85=0.1*x+4*x^2 equation: 


Simplifying
18.85 = 0.1x + 4x2

Solving
18.85 = 0.1x + 4x2

Solving for variable 'x'.

Reorder the terms:
18.85 + -0.1x + -4x2 = 0.1x + -0.1x + 4x2 + -4x2

Combine like terms: 0.1x + -0.1x = 0.0
18.85 + -0.1x + -4x2 = 0.0 + 4x2 + -4x2
18.85 + -0.1x + -4x2 = 4x2 + -4x2

Combine like terms: 4x2 + -4x2 = 0
18.85 + -0.1x + -4x2 = 0

Begin completing the square.  Divide all terms by
-4 the coefficient of the squared term: 

Divide each side by '-4'.
-4.7125 + 0.025x + x2 = 0

Move the constant term to the right:

Add '4.7125' to each side of the equation.
-4.7125 + 0.025x + 4.7125 + x2 = 0 + 4.7125

Reorder the terms:
-4.7125 + 4.7125 + 0.025x + x2 = 0 + 4.7125

Combine like terms: -4.7125 + 4.7125 = 0.0000
0.0000 + 0.025x + x2 = 0 + 4.7125
0.025x + x2 = 0 + 4.7125

Combine like terms: 0 + 4.7125 = 4.7125
0.025x + x2 = 4.7125

The x term is 0.025x.  Take half its coefficient (0.0125).
Square it (0.00015625) and add it to both sides.

Add '0.00015625' to each side of the equation.
0.025x + 0.00015625 + x2 = 4.7125 + 0.00015625

Reorder the terms:
0.00015625 + 0.025x + x2 = 4.7125 + 0.00015625

Combine like terms: 4.7125 + 0.00015625 = 4.71265625
0.00015625 + 0.025x + x2 = 4.71265625

Factor a perfect square on the left side:
(x + 0.0125)(x + 0.0125) = 4.71265625

Calculate the square root of the right side: 2.170865323

Break this problem into two subproblems by setting 
(x + 0.0125) equal to 2.170865323 and -2.170865323.


Subproblem 1
x + 0.0125 = 2.170865323

Simplifying
x + 0.0125 = 2.170865323

Reorder the terms:
0.0125 + x = 2.170865323

Solving
0.0125 + x = 2.170865323

Solving for variable 'x'.

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

Add '-0.0125' to each side of the equation.
0.0125 + -0.0125 + x = 2.170865323 + -0.0125

Combine like terms: 0.0125 + -0.0125 = 0.0000
0.0000 + x = 2.170865323 + -0.0125
x = 2.170865323 + -0.0125

Combine like terms: 2.170865323 + -0.0125 = 2.158365323
x = 2.158365323

Simplifying
x = 2.158365323


Subproblem 2
x + 0.0125 = -2.170865323

Simplifying
x + 0.0125 = -2.170865323

Reorder the terms:
0.0125 + x = -2.170865323

Solving
0.0125 + x = -2.170865323

Solving for variable 'x'.

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

Add '-0.0125' to each side of the equation.
0.0125 + -0.0125 + x = -2.170865323 + -0.0125

Combine like terms: 0.0125 + -0.0125 = 0.0000
0.0000 + x = -2.170865323 + -0.0125
x = -2.170865323 + -0.0125

Combine like terms: -2.170865323 + -0.0125 = -2.183365323
x = -2.183365323

Simplifying
x = -2.183365323


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.158365323, -2.183365323}

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