r^2+4r+1=r

Solution for r^2+4r+1=r equation:

Simplifying
r2 + 4r + 1 = r

Reorder the terms:
1 + 4r + r2 = r

Solving
1 + 4r + r2 = r

Solving for variable 'r'.

Reorder the terms:
1 + 4r + -1r + r2 = r + -1r

Combine like terms: 4r + -1r = 3r
1 + 3r + r2 = r + -1r

Combine like terms: r + -1r = 0
1 + 3r + r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 3r + -1 + r2 = 0 + -1

Reorder the terms:
1 + -1 + 3r + r2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 3r + r2 = 0 + -1
3r + r2 = 0 + -1

Combine like terms: 0 + -1 = -1
3r + r2 = -1

The r term is 3r.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3r + 2.25 + r2 = -1 + 2.25

Reorder the terms:
2.25 + 3r + r2 = -1 + 2.25

Combine like terms: -1 + 2.25 = 1.25
2.25 + 3r + r2 = 1.25

Factor a perfect square on the left side:
(r + 1.5)(r + 1.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(r + 1.5) equal to 1.118033989 and -1.118033989.

Subproblem 1
r + 1.5 = 1.118033989

Simplifying
r + 1.5 = 1.118033989

Reorder the terms:
1.5 + r = 1.118033989

Solving
1.5 + r = 1.118033989

Solving for variable 'r'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + r = 1.118033989 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + r = 1.118033989 + -1.5
r = 1.118033989 + -1.5

Combine like terms: 1.118033989 + -1.5 = -0.381966011
r = -0.381966011

Simplifying
r = -0.381966011

Subproblem 2
r + 1.5 = -1.118033989

Simplifying
r + 1.5 = -1.118033989

Reorder the terms:
1.5 + r = -1.118033989

Solving
1.5 + r = -1.118033989

Solving for variable 'r'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + r = -1.118033989 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + r = -1.118033989 + -1.5
r = -1.118033989 + -1.5

Combine like terms: -1.118033989 + -1.5 = -2.618033989
r = -2.618033989

Simplifying
r = -2.618033989

Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {-0.381966011, -2.618033989}