m^2+5m+1=0

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

Simplifying
m2 + 5m + 1 = 0

Reorder the terms:
1 + 5m + m2 = 0

Solving
1 + 5m + m2 = 0

Solving for variable 'm'.

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 5m + -1 + m2 = 0 + -1

Reorder the terms:
1 + -1 + 5m + m2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 5m + m2 = 0 + -1
5m + m2 = 0 + -1

Combine like terms: 0 + -1 = -1
5m + m2 = -1

The m term is 5m.  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.
5m + 6.25 + m2 = -1 + 6.25

Reorder the terms:
6.25 + 5m + m2 = -1 + 6.25

Combine like terms: -1 + 6.25 = 5.25
6.25 + 5m + m2 = 5.25

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

Calculate the square root of the right side: 2.291287847

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

Subproblem 1
m + 2.5 = 2.291287847

Simplifying
m + 2.5 = 2.291287847

Reorder the terms:
2.5 + m = 2.291287847

Solving
2.5 + m = 2.291287847

Solving for variable 'm'.

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

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

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

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

Simplifying
m = -0.208712153

Subproblem 2
m + 2.5 = -2.291287847

Simplifying
m + 2.5 = -2.291287847

Reorder the terms:
2.5 + m = -2.291287847

Solving
2.5 + m = -2.291287847

Solving for variable 'm'.

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

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

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

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

Simplifying
m = -4.791287847

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