x+x*x=394

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Solution for x+x*x=394 equation: 


Simplifying
x + x * x = 394

Multiply x * x
x + x2 = 394

Solving
x + x2 = 394

Solving for variable 'x'.

Reorder the terms:
-394 + x + x2 = 394 + -394

Combine like terms: 394 + -394 = 0
-394 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '394' to each side of the equation.
-394 + x + 394 + x2 = 0 + 394

Reorder the terms:
-394 + 394 + x + x2 = 0 + 394

Combine like terms: -394 + 394 = 0
0 + x + x2 = 0 + 394
x + x2 = 0 + 394

Combine like terms: 0 + 394 = 394
x + x2 = 394

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.
x + 0.25 + x2 = 394 + 0.25

Reorder the terms:
0.25 + x + x2 = 394 + 0.25

Combine like terms: 394 + 0.25 = 394.25
0.25 + x + x2 = 394.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 394.25

Calculate the square root of the right side: 19.855729652

Break this problem into two subproblems by setting 
(x + 0.5) equal to 19.855729652 and -19.855729652.


Subproblem 1
x + 0.5 = 19.855729652

Simplifying
x + 0.5 = 19.855729652

Reorder the terms:
0.5 + x = 19.855729652

Solving
0.5 + x = 19.855729652

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 = 19.855729652 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 19.855729652 + -0.5
x = 19.855729652 + -0.5

Combine like terms: 19.855729652 + -0.5 = 19.355729652
x = 19.355729652

Simplifying
x = 19.355729652


Subproblem 2
x + 0.5 = -19.855729652

Simplifying
x + 0.5 = -19.855729652

Reorder the terms:
0.5 + x = -19.855729652

Solving
0.5 + x = -19.855729652

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 = -19.855729652 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -19.855729652 + -0.5
x = -19.855729652 + -0.5

Combine like terms: -19.855729652 + -0.5 = -20.355729652
x = -20.355729652

Simplifying
x = -20.355729652


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {19.355729652, -20.355729652}

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