r^2+5r+1=0

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Solution for r^2+5r+1=0 equation: 


Simplifying
r2 + 5r + 1 = 0

Reorder the terms:
1 + 5r + r2 = 0

Solving
1 + 5r + r2 = 0

Solving for variable 'r'.

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 5r + -1 + r2 = 0 + -1

Reorder the terms:
1 + -1 + 5r + r2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 5r + r2 = 0 + -1
5r + r2 = 0 + -1

Combine like terms: 0 + -1 = -1
5r + r2 = -1

The r term is 5r.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5r + 6.25 + r2 = -1 + 6.25

Reorder the terms:
6.25 + 5r + r2 = -1 + 6.25

Combine like terms: -1 + 6.25 = 5.25
6.25 + 5r + r2 = 5.25

Factor a perfect square on the left side:
(r + 2.5)(r + 2.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(r + 2.5) equal to 2.291287847 and -2.291287847.


Subproblem 1
r + 2.5 = 2.291287847

Simplifying
r + 2.5 = 2.291287847

Reorder the terms:
2.5 + r = 2.291287847

Solving
2.5 + r = 2.291287847

Solving for variable 'r'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + r = 2.291287847 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + r = 2.291287847 + -2.5
r = 2.291287847 + -2.5

Combine like terms: 2.291287847 + -2.5 = -0.208712153
r = -0.208712153

Simplifying
r = -0.208712153


Subproblem 2
r + 2.5 = -2.291287847

Simplifying
r + 2.5 = -2.291287847

Reorder the terms:
2.5 + r = -2.291287847

Solving
2.5 + r = -2.291287847

Solving for variable 'r'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + r = -2.291287847 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + r = -2.291287847 + -2.5
r = -2.291287847 + -2.5

Combine like terms: -2.291287847 + -2.5 = -4.791287847
r = -4.791287847

Simplifying
r = -4.791287847


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {-0.208712153, -4.791287847}

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