2r+(2r*2r)=84

Solution for 2r+(2r*2r)=84 equation:

Simplifying
2r + (2r * 2r) = 84

Reorder the terms for easier multiplication:
2r + (2 * 2r * r) = 84

Multiply 2 * 2
2r + (4r * r) = 84

Multiply r * r
2r + (4r2) = 84

Solving
2r + (4r2) = 84

Solving for variable 'r'.

Reorder the terms:
-84 + 2r + (4r2) = 84 + -84

Combine like terms: 84 + -84 = 0
-84 + 2r + (4r2) = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-42 + r + (2r2)) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-42 + r + (2r2))' equal to zero and attempt to solve:

Simplifying
-42 + r + (2r2) = 0

Solving
-42 + r + (2r2) = 0

Begin completing the square.  Divide all terms by
2 the coefficient of the squared term: 

Divide each side by '2'.
-21 + 0.5r + r2 = 0

Move the constant term to the right:

Add '21' to each side of the equation.
-21 + 0.5r + 21 + r2 = 0 + 21

Reorder the terms:
-21 + 21 + 0.5r + r2 = 0 + 21

Combine like terms: -21 + 21 = 0
0 + 0.5r + r2 = 0 + 21
0.5r + r2 = 0 + 21

Combine like terms: 0 + 21 = 21
0.5r + r2 = 21

The r term is r.  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.
0.5r + 0.25 + r2 = 21 + 0.25

Reorder the terms:
0.25 + 0.5r + r2 = 21 + 0.25

Combine like terms: 21 + 0.25 = 21.25
0.25 + 0.5r + 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
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
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}
Solution
r = {4.109772229, -5.109772229}