x(x+1)=399

Solution for x(x+1)=399 equation:

Simplifying
x(x + 1) = 399

Reorder the terms:
x(1 + x) = 399
(1 * x + x * x) = 399
(1x + x2) = 399

Solving
1x + x2 = 399

Solving for variable 'x'.

Reorder the terms:
-399 + 1x + x2 = 399 + -399

Combine like terms: 399 + -399 = 0
-399 + 1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '399' to each side of the equation.
-399 + 1x + 399 + x2 = 0 + 399

Reorder the terms:
-399 + 399 + 1x + x2 = 0 + 399

Combine like terms: -399 + 399 = 0
0 + 1x + x2 = 0 + 399
1x + x2 = 0 + 399

Combine like terms: 0 + 399 = 399
1x + x2 = 399

The x term is 1x.  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 = 399 + 0.25

Reorder the terms:
0.25 + x + x2 = 399 + 0.25

Combine like terms: 399 + 0.25 = 399.25
0.25 + x + x2 = 399.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 399.25

Calculate the square root of the right side: 19.981241203

Break this problem into two subproblems by setting 
(x + 0.5) equal to 19.981241203 and -19.981241203.

Subproblem 1
x + 0.5 = 19.981241203

Simplifying
x + 0.5 = 19.981241203

Reorder the terms:
0.5 + x = 19.981241203

Solving
0.5 + x = 19.981241203

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 = 19.981241203 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 19.981241203 + -0.5
x = 19.981241203 + -0.5

Combine like terms: 19.981241203 + -0.5 = 19.481241203
x = 19.481241203

Simplifying
x = 19.481241203

Subproblem 2
x + 0.5 = -19.981241203

Simplifying
x + 0.5 = -19.981241203

Reorder the terms:
0.5 + x = -19.981241203

Solving
0.5 + x = -19.981241203

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 = -19.981241203 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -19.981241203 + -0.5
x = -19.981241203 + -0.5

Combine like terms: -19.981241203 + -0.5 = -20.481241203
x = -20.481241203

Simplifying
x = -20.481241203

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {19.481241203, -20.481241203}