2u^2+5u+1=0

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

Simplifying
2u2 + 5u + 1 = 0

Reorder the terms:
1 + 5u + 2u2 = 0

Solving
1 + 5u + 2u2 = 0

Solving for variable 'u'.

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

Divide each side by '2'.
0.5 + 2.5u + u2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + 2.5u + -0.5 + u2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + 2.5u + u2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + 2.5u + u2 = 0 + -0.5
2.5u + u2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
2.5u + u2 = -0.5

The u term is 2.5u.  Take half its coefficient (1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
2.5u + 1.5625 + u2 = -0.5 + 1.5625

Reorder the terms:
1.5625 + 2.5u + u2 = -0.5 + 1.5625

Combine like terms: -0.5 + 1.5625 = 1.0625
1.5625 + 2.5u + u2 = 1.0625

Factor a perfect square on the left side:
(u + 1.25)(u + 1.25) = 1.0625

Calculate the square root of the right side: 1.030776406

Break this problem into two subproblems by setting 
(u + 1.25) equal to 1.030776406 and -1.030776406.

Subproblem 1
u + 1.25 = 1.030776406

Simplifying
u + 1.25 = 1.030776406

Reorder the terms:
1.25 + u = 1.030776406

Solving
1.25 + u = 1.030776406

Solving for variable 'u'.

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

Add '-1.25' to each side of the equation.
1.25 + -1.25 + u = 1.030776406 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + u = 1.030776406 + -1.25
u = 1.030776406 + -1.25

Combine like terms: 1.030776406 + -1.25 = -0.219223594
u = -0.219223594

Simplifying
u = -0.219223594

Subproblem 2
u + 1.25 = -1.030776406

Simplifying
u + 1.25 = -1.030776406

Reorder the terms:
1.25 + u = -1.030776406

Solving
1.25 + u = -1.030776406

Solving for variable 'u'.

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

Add '-1.25' to each side of the equation.
1.25 + -1.25 + u = -1.030776406 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + u = -1.030776406 + -1.25
u = -1.030776406 + -1.25

Combine like terms: -1.030776406 + -1.25 = -2.280776406
u = -2.280776406

Simplifying
u = -2.280776406

Solution
The solution to the problem is based on the solutions
from the subproblems.
u = {-0.219223594, -2.280776406}