x(x+1)=1024

Solution for x(x+1)=1024 equation:

Simplifying
x(x + 1) = 1024

Reorder the terms:
x(1 + x) = 1024
(1 * x + x * x) = 1024
(1x + x2) = 1024

Solving
1x + x2 = 1024

Solving for variable 'x'.

Reorder the terms:
-1024 + 1x + x2 = 1024 + -1024

Combine like terms: 1024 + -1024 = 0
-1024 + 1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1024' to each side of the equation.
-1024 + 1x + 1024 + x2 = 0 + 1024

Reorder the terms:
-1024 + 1024 + 1x + x2 = 0 + 1024

Combine like terms: -1024 + 1024 = 0
0 + 1x + x2 = 0 + 1024
1x + x2 = 0 + 1024

Combine like terms: 0 + 1024 = 1024
1x + x2 = 1024

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 = 1024 + 0.25

Reorder the terms:
0.25 + x + x2 = 1024 + 0.25

Combine like terms: 1024 + 0.25 = 1024.25
0.25 + x + x2 = 1024.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 1024.25

Calculate the square root of the right side: 32.003906012

Break this problem into two subproblems by setting 
(x + 0.5) equal to 32.003906012 and -32.003906012.

Subproblem 1
x + 0.5 = 32.003906012

Simplifying
x + 0.5 = 32.003906012

Reorder the terms:
0.5 + x = 32.003906012

Solving
0.5 + x = 32.003906012

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 = 32.003906012 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 32.003906012 + -0.5
x = 32.003906012 + -0.5

Combine like terms: 32.003906012 + -0.5 = 31.503906012
x = 31.503906012

Simplifying
x = 31.503906012

Subproblem 2
x + 0.5 = -32.003906012

Simplifying
x + 0.5 = -32.003906012

Reorder the terms:
0.5 + x = -32.003906012

Solving
0.5 + x = -32.003906012

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 = -32.003906012 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -32.003906012 + -0.5
x = -32.003906012 + -0.5

Combine like terms: -32.003906012 + -0.5 = -32.503906012
x = -32.503906012

Simplifying
x = -32.503906012

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {31.503906012, -32.503906012}