4z^2-20z+1=0

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Solution for 4z^2-20z+1=0 equation: 


Simplifying
4z2 + -20z + 1 = 0

Reorder the terms:
1 + -20z + 4z2 = 0

Solving
1 + -20z + 4z2 = 0

Solving for variable 'z'.

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

Divide each side by '4'.
0.25 + -5z + z2 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + -5z + -0.25 + z2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + -5z + z2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + -5z + z2 = 0 + -0.25
-5z + z2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
-5z + z2 = -0.25

The z term is -5z.  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.
-5z + 6.25 + z2 = -0.25 + 6.25

Reorder the terms:
6.25 + -5z + z2 = -0.25 + 6.25

Combine like terms: -0.25 + 6.25 = 6
6.25 + -5z + z2 = 6

Factor a perfect square on the left side:
(z + -2.5)(z + -2.5) = 6

Calculate the square root of the right side: 2.449489743

Break this problem into two subproblems by setting 
(z + -2.5) equal to 2.449489743 and -2.449489743.


Subproblem 1
z + -2.5 = 2.449489743

Simplifying
z + -2.5 = 2.449489743

Reorder the terms:
-2.5 + z = 2.449489743

Solving
-2.5 + z = 2.449489743

Solving for variable 'z'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + z = 2.449489743 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + z = 2.449489743 + 2.5
z = 2.449489743 + 2.5

Combine like terms: 2.449489743 + 2.5 = 4.949489743
z = 4.949489743

Simplifying
z = 4.949489743


Subproblem 2
z + -2.5 = -2.449489743

Simplifying
z + -2.5 = -2.449489743

Reorder the terms:
-2.5 + z = -2.449489743

Solving
-2.5 + z = -2.449489743

Solving for variable 'z'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + z = -2.449489743 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + z = -2.449489743 + 2.5
z = -2.449489743 + 2.5

Combine like terms: -2.449489743 + 2.5 = 0.050510257
z = 0.050510257

Simplifying
z = 0.050510257


Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {4.949489743, 0.050510257}

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