r^2+5r+2=0

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


Simplifying
r2 + 5r + 2 = 0

Reorder the terms:
2 + 5r + r2 = 0

Solving
2 + 5r + r2 = 0

Solving for variable 'r'.

Begin completing the square.

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + 5r + -2 + r2 = 0 + -2

Reorder the terms:
2 + -2 + 5r + r2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + 5r + r2 = 0 + -2
5r + r2 = 0 + -2

Combine like terms: 0 + -2 = -2
5r + r2 = -2

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 = -2 + 6.25

Reorder the terms:
6.25 + 5r + r2 = -2 + 6.25

Combine like terms: -2 + 6.25 = 4.25
6.25 + 5r + r2 = 4.25

Factor a perfect square on the left side:
(r + 2.5)(r + 2.5) = 4.25

Calculate the square root of the right side: 2.061552813

Break this problem into two subproblems by setting 
(r + 2.5) equal to 2.061552813 and -2.061552813.


Subproblem 1
r + 2.5 = 2.061552813

Simplifying
r + 2.5 = 2.061552813

Reorder the terms:
2.5 + r = 2.061552813

Solving
2.5 + r = 2.061552813

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 = 2.061552813 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + r = 2.061552813 + -2.5
r = 2.061552813 + -2.5

Combine like terms: 2.061552813 + -2.5 = -0.438447187
r = -0.438447187

Simplifying
r = -0.438447187


Subproblem 2
r + 2.5 = -2.061552813

Simplifying
r + 2.5 = -2.061552813

Reorder the terms:
2.5 + r = -2.061552813

Solving
2.5 + r = -2.061552813

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 = -2.061552813 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + r = -2.061552813 + -2.5
r = -2.061552813 + -2.5

Combine like terms: -2.061552813 + -2.5 = -4.561552813
r = -4.561552813

Simplifying
r = -4.561552813


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {-0.438447187, -4.561552813}

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