x(x+1)=2400

Solution for x(x+1)=2400 equation:

Simplifying
x(x + 1) = 2400

Reorder the terms:
x(1 + x) = 2400
(1 * x + x * x) = 2400
(1x + x2) = 2400

Solving
1x + x2 = 2400

Solving for variable 'x'.

Reorder the terms:
-2400 + 1x + x2 = 2400 + -2400

Combine like terms: 2400 + -2400 = 0
-2400 + 1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '2400' to each side of the equation.
-2400 + 1x + 2400 + x2 = 0 + 2400

Reorder the terms:
-2400 + 2400 + 1x + x2 = 0 + 2400

Combine like terms: -2400 + 2400 = 0
0 + 1x + x2 = 0 + 2400
1x + x2 = 0 + 2400

Combine like terms: 0 + 2400 = 2400
1x + x2 = 2400

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 = 2400 + 0.25

Reorder the terms:
0.25 + x + x2 = 2400 + 0.25

Combine like terms: 2400 + 0.25 = 2400.25
0.25 + x + x2 = 2400.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 2400.25

Calculate the square root of the right side: 48.992346341

Break this problem into two subproblems by setting 
(x + 0.5) equal to 48.992346341 and -48.992346341.

Subproblem 1
x + 0.5 = 48.992346341

Simplifying
x + 0.5 = 48.992346341

Reorder the terms:
0.5 + x = 48.992346341

Solving
0.5 + x = 48.992346341

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 = 48.992346341 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 48.992346341 + -0.5
x = 48.992346341 + -0.5

Combine like terms: 48.992346341 + -0.5 = 48.492346341
x = 48.492346341

Simplifying
x = 48.492346341

Subproblem 2
x + 0.5 = -48.992346341

Simplifying
x + 0.5 = -48.992346341

Reorder the terms:
0.5 + x = -48.992346341

Solving
0.5 + x = -48.992346341

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 = -48.992346341 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -48.992346341 + -0.5
x = -48.992346341 + -0.5

Combine like terms: -48.992346341 + -0.5 = -49.492346341
x = -49.492346341

Simplifying
x = -49.492346341

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {48.492346341, -49.492346341}