r(r+1)=21

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

Simplifying
r(r + 1) = 21

Reorder the terms:
r(1 + r) = 21
(1 * r + r * r) = 21
(1r + r2) = 21

Solving
1r + r2 = 21

Solving for variable 'r'.

Reorder the terms:
-21 + 1r + r2 = 21 + -21

Combine like terms: 21 + -21 = 0
-21 + 1r + r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '21' to each side of the equation.
-21 + 1r + 21 + r2 = 0 + 21

Reorder the terms:
-21 + 21 + 1r + r2 = 0 + 21

Combine like terms: -21 + 21 = 0
0 + 1r + r2 = 0 + 21
1r + r2 = 0 + 21

Combine like terms: 0 + 21 = 21
1r + r2 = 21

The r term is 1r.  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.
r + 0.25 + r2 = 21 + 0.25

Reorder the terms:
0.25 + r + r2 = 21 + 0.25

Combine like terms: 21 + 0.25 = 21.25
0.25 + r + r2 = 21.25

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

Calculate the square root of the right side: 4.609772229

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

Subproblem 1
r + 0.5 = 4.609772229

Simplifying
r + 0.5 = 4.609772229

Reorder the terms:
0.5 + r = 4.609772229

Solving
0.5 + r = 4.609772229

Solving for variable 'r'.

Move all terms containing r to the left, all other terms to the right.

Add '-0.5' to each side of the equation.
0.5 + -0.5 + r = 4.609772229 + -0.5

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

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

Simplifying
r = 4.109772229

Subproblem 2
r + 0.5 = -4.609772229

Simplifying
r + 0.5 = -4.609772229

Reorder the terms:
0.5 + r = -4.609772229

Solving
0.5 + r = -4.609772229

Solving for variable 'r'.

Move all terms containing r to the left, all other terms to the right.

Add '-0.5' to each side of the equation.
0.5 + -0.5 + r = -4.609772229 + -0.5

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

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

Simplifying
r = -5.109772229

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