10z^2-23z+2=0

Solution for 10z^2-23z+2=0 equation:

Simplifying
10z2 + -23z + 2 = 0

Reorder the terms:
2 + -23z + 10z2 = 0

Solving
2 + -23z + 10z2 = 0

Solving for variable 'z'.

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

Divide each side by '10'.
0.2 + -2.3z + z2 = 0

Move the constant term to the right:

Add '-0.2' to each side of the equation.
0.2 + -2.3z + -0.2 + z2 = 0 + -0.2

Reorder the terms:
0.2 + -0.2 + -2.3z + z2 = 0 + -0.2

Combine like terms: 0.2 + -0.2 = 0.0
0.0 + -2.3z + z2 = 0 + -0.2
-2.3z + z2 = 0 + -0.2

Combine like terms: 0 + -0.2 = -0.2
-2.3z + z2 = -0.2

The z term is -2.3z.  Take half its coefficient (-1.15).
Square it (1.3225) and add it to both sides.

Add '1.3225' to each side of the equation.
-2.3z + 1.3225 + z2 = -0.2 + 1.3225

Reorder the terms:
1.3225 + -2.3z + z2 = -0.2 + 1.3225

Combine like terms: -0.2 + 1.3225 = 1.1225
1.3225 + -2.3z + z2 = 1.1225

Factor a perfect square on the left side:
(z + -1.15)(z + -1.15) = 1.1225

Calculate the square root of the right side: 1.059481005

Break this problem into two subproblems by setting 
(z + -1.15) equal to 1.059481005 and -1.059481005.

Subproblem 1
z + -1.15 = 1.059481005

Simplifying
z + -1.15 = 1.059481005

Reorder the terms:
-1.15 + z = 1.059481005

Solving
-1.15 + z = 1.059481005

Solving for variable 'z'.

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

Add '1.15' to each side of the equation.
-1.15 + 1.15 + z = 1.059481005 + 1.15

Combine like terms: -1.15 + 1.15 = 0.00
0.00 + z = 1.059481005 + 1.15
z = 1.059481005 + 1.15

Combine like terms: 1.059481005 + 1.15 = 2.209481005
z = 2.209481005

Simplifying
z = 2.209481005

Subproblem 2
z + -1.15 = -1.059481005

Simplifying
z + -1.15 = -1.059481005

Reorder the terms:
-1.15 + z = -1.059481005

Solving
-1.15 + z = -1.059481005

Solving for variable 'z'.

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

Add '1.15' to each side of the equation.
-1.15 + 1.15 + z = -1.059481005 + 1.15

Combine like terms: -1.15 + 1.15 = 0.00
0.00 + z = -1.059481005 + 1.15
z = -1.059481005 + 1.15

Combine like terms: -1.059481005 + 1.15 = 0.090518995
z = 0.090518995

Simplifying
z = 0.090518995

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {2.209481005, 0.090518995}