2z^2-5z+1=0

Solution for 2z^2-5z+1=0 equation:

Simplifying
2z2 + -5z + 1 = 0

Reorder the terms:
1 + -5z + 2z2 = 0

Solving
1 + -5z + 2z2 = 0

Solving for variable 'z'.

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

Divide each side by '2'.
0.5 + -2.5z + z2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + -2.5z + -0.5 + z2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + -2.5z + z2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + -2.5z + z2 = 0 + -0.5
-2.5z + z2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
-2.5z + z2 = -0.5

The z term is -2.5z.  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.
-2.5z + 1.5625 + z2 = -0.5 + 1.5625

Reorder the terms:
1.5625 + -2.5z + z2 = -0.5 + 1.5625

Combine like terms: -0.5 + 1.5625 = 1.0625
1.5625 + -2.5z + z2 = 1.0625

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

Calculate the square root of the right side: 1.030776406

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

Subproblem 1
z + -1.25 = 1.030776406

Simplifying
z + -1.25 = 1.030776406

Reorder the terms:
-1.25 + z = 1.030776406

Solving
-1.25 + z = 1.030776406

Solving for variable 'z'.

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

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

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

Combine like terms: 1.030776406 + 1.25 = 2.280776406
z = 2.280776406

Simplifying
z = 2.280776406

Subproblem 2
z + -1.25 = -1.030776406

Simplifying
z + -1.25 = -1.030776406

Reorder the terms:
-1.25 + z = -1.030776406

Solving
-1.25 + z = -1.030776406

Solving for variable 'z'.

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

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

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

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

Simplifying
z = 0.219223594

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