x(x+1)=21

Solution for x(x+1)=21 equation:

Simplifying
x(x + 1) = 21

Reorder the terms:
x(1 + x) = 21
(1 * x + x * x) = 21
(1x + x2) = 21

Solving
1x + x2 = 21

Solving for variable 'x'.

Reorder the terms:
-21 + 1x + x2 = 21 + -21

Combine like terms: 21 + -21 = 0
-21 + 1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '21' to each side of the equation.
-21 + 1x + 21 + x2 = 0 + 21

Reorder the terms:
-21 + 21 + 1x + x2 = 0 + 21

Combine like terms: -21 + 21 = 0
0 + 1x + x2 = 0 + 21
1x + x2 = 0 + 21

Combine like terms: 0 + 21 = 21
1x + x2 = 21

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 = 21 + 0.25

Reorder the terms:
0.25 + x + x2 = 21 + 0.25

Combine like terms: 21 + 0.25 = 21.25
0.25 + x + x2 = 21.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 21.25

Calculate the square root of the right side: 4.609772229

Break this problem into two subproblems by setting 
(x + 0.5) equal to 4.609772229 and -4.609772229.

Subproblem 1
x + 0.5 = 4.609772229

Simplifying
x + 0.5 = 4.609772229

Reorder the terms:
0.5 + x = 4.609772229

Solving
0.5 + x = 4.609772229

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 = 4.609772229 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 4.609772229 + -0.5
x = 4.609772229 + -0.5

Combine like terms: 4.609772229 + -0.5 = 4.109772229
x = 4.109772229

Simplifying
x = 4.109772229

Subproblem 2
x + 0.5 = -4.609772229

Simplifying
x + 0.5 = -4.609772229

Reorder the terms:
0.5 + x = -4.609772229

Solving
0.5 + x = -4.609772229

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 = -4.609772229 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -4.609772229 + -0.5
x = -4.609772229 + -0.5

Combine like terms: -4.609772229 + -0.5 = -5.109772229
x = -5.109772229

Simplifying
x = -5.109772229

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.109772229, -5.109772229}