-5z^2-6z+4=0

Solution for -5z^2-6z+4=0 equation:

Simplifying
-5z2 + -6z + 4 = 0

Reorder the terms:
4 + -6z + -5z2 = 0

Solving
4 + -6z + -5z2 = 0

Solving for variable 'z'.

Begin completing the square.  Divide all terms by
-5 the coefficient of the squared term: 

Divide each side by '-5'.
-0.8 + 1.2z + z2 = 0

Move the constant term to the right:

Add '0.8' to each side of the equation.
-0.8 + 1.2z + 0.8 + z2 = 0 + 0.8

Reorder the terms:
-0.8 + 0.8 + 1.2z + z2 = 0 + 0.8

Combine like terms: -0.8 + 0.8 = 0.0
0.0 + 1.2z + z2 = 0 + 0.8
1.2z + z2 = 0 + 0.8

Combine like terms: 0 + 0.8 = 0.8
1.2z + z2 = 0.8

The z term is 1.2z.  Take half its coefficient (0.6).
Square it (0.36) and add it to both sides.

Add '0.36' to each side of the equation.
1.2z + 0.36 + z2 = 0.8 + 0.36

Reorder the terms:
0.36 + 1.2z + z2 = 0.8 + 0.36

Combine like terms: 0.8 + 0.36 = 1.16
0.36 + 1.2z + z2 = 1.16

Factor a perfect square on the left side:
(z + 0.6)(z + 0.6) = 1.16

Calculate the square root of the right side: 1.077032961

Break this problem into two subproblems by setting 
(z + 0.6) equal to 1.077032961 and -1.077032961.

Subproblem 1
z + 0.6 = 1.077032961

Simplifying
z + 0.6 = 1.077032961

Reorder the terms:
0.6 + z = 1.077032961

Solving
0.6 + z = 1.077032961

Solving for variable 'z'.

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

Add '-0.6' to each side of the equation.
0.6 + -0.6 + z = 1.077032961 + -0.6

Combine like terms: 0.6 + -0.6 = 0.0
0.0 + z = 1.077032961 + -0.6
z = 1.077032961 + -0.6

Combine like terms: 1.077032961 + -0.6 = 0.477032961
z = 0.477032961

Simplifying
z = 0.477032961

Subproblem 2
z + 0.6 = -1.077032961

Simplifying
z + 0.6 = -1.077032961

Reorder the terms:
0.6 + z = -1.077032961

Solving
0.6 + z = -1.077032961

Solving for variable 'z'.

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

Add '-0.6' to each side of the equation.
0.6 + -0.6 + z = -1.077032961 + -0.6

Combine like terms: 0.6 + -0.6 = 0.0
0.0 + z = -1.077032961 + -0.6
z = -1.077032961 + -0.6

Combine like terms: -1.077032961 + -0.6 = -1.677032961
z = -1.677032961

Simplifying
z = -1.677032961

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {0.477032961, -1.677032961}