x(x+1)=511

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Solution for x(x+1)=511 equation: 


Simplifying
x(x + 1) = 511

Reorder the terms:
x(1 + x) = 511
(1 * x + x * x) = 511
(1x + x2) = 511

Solving
1x + x2 = 511

Solving for variable 'x'.

Reorder the terms:
-511 + 1x + x2 = 511 + -511

Combine like terms: 511 + -511 = 0
-511 + 1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '511' to each side of the equation.
-511 + 1x + 511 + x2 = 0 + 511

Reorder the terms:
-511 + 511 + 1x + x2 = 0 + 511

Combine like terms: -511 + 511 = 0
0 + 1x + x2 = 0 + 511
1x + x2 = 0 + 511

Combine like terms: 0 + 511 = 511
1x + x2 = 511

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 = 511 + 0.25

Reorder the terms:
0.25 + x + x2 = 511 + 0.25

Combine like terms: 511 + 0.25 = 511.25
0.25 + x + x2 = 511.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 511.25

Calculate the square root of the right side: 22.610838109

Break this problem into two subproblems by setting 
(x + 0.5) equal to 22.610838109 and -22.610838109.


Subproblem 1
x + 0.5 = 22.610838109

Simplifying
x + 0.5 = 22.610838109

Reorder the terms:
0.5 + x = 22.610838109

Solving
0.5 + x = 22.610838109

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 = 22.610838109 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 22.610838109 + -0.5
x = 22.610838109 + -0.5

Combine like terms: 22.610838109 + -0.5 = 22.110838109
x = 22.110838109

Simplifying
x = 22.110838109


Subproblem 2
x + 0.5 = -22.610838109

Simplifying
x + 0.5 = -22.610838109

Reorder the terms:
0.5 + x = -22.610838109

Solving
0.5 + x = -22.610838109

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 = -22.610838109 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -22.610838109 + -0.5
x = -22.610838109 + -0.5

Combine like terms: -22.610838109 + -0.5 = -23.110838109
x = -23.110838109

Simplifying
x = -23.110838109


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {22.110838109, -23.110838109}

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