z(4z-1)=3(z+2)

Solution for z(4z-1)=3(z+2) equation:

Simplifying
z(4z + -1) = 3(z + 2)

Reorder the terms:
z(-1 + 4z) = 3(z + 2)
(-1 * z + 4z * z) = 3(z + 2)
(-1z + 4z2) = 3(z + 2)

Reorder the terms:
-1z + 4z2 = 3(2 + z)
-1z + 4z2 = (2 * 3 + z * 3)
-1z + 4z2 = (6 + 3z)

Solving
-1z + 4z2 = 6 + 3z

Solving for variable 'z'.

Reorder the terms:
-6 + -1z + -3z + 4z2 = 6 + 3z + -6 + -3z

Combine like terms: -1z + -3z = -4z
-6 + -4z + 4z2 = 6 + 3z + -6 + -3z

Reorder the terms:
-6 + -4z + 4z2 = 6 + -6 + 3z + -3z

Combine like terms: 6 + -6 = 0
-6 + -4z + 4z2 = 0 + 3z + -3z
-6 + -4z + 4z2 = 3z + -3z

Combine like terms: 3z + -3z = 0
-6 + -4z + 4z2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-3 + -2z + 2z2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-3 + -2z + 2z2)' equal to zero and attempt to solve:

Simplifying
-3 + -2z + 2z2 = 0

Solving
-3 + -2z + 2z2 = 0

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

Divide each side by '2'.
-1.5 + -1z + z2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + -1z + 1.5 + z2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + -1z + z2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + -1z + z2 = 0 + 1.5
-1z + z2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
-1z + z2 = 1.5

The z term is -1z.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1z + 0.25 + z2 = 1.5 + 0.25

Reorder the terms:
0.25 + -1z + z2 = 1.5 + 0.25

Combine like terms: 1.5 + 0.25 = 1.75
0.25 + -1z + z2 = 1.75

Factor a perfect square on the left side:
(z + -0.5)(z + -0.5) = 1.75

Calculate the square root of the right side: 1.322875656

Break this problem into two subproblems by setting 
(z + -0.5) equal to 1.322875656 and -1.322875656.

Subproblem 1
z + -0.5 = 1.322875656

Simplifying
z + -0.5 = 1.322875656

Reorder the terms:
-0.5 + z = 1.322875656

Solving
-0.5 + z = 1.322875656

Solving for variable 'z'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + z = 1.322875656 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + z = 1.322875656 + 0.5
z = 1.322875656 + 0.5

Combine like terms: 1.322875656 + 0.5 = 1.822875656
z = 1.822875656

Simplifying
z = 1.822875656

Subproblem 2
z + -0.5 = -1.322875656

Simplifying
z + -0.5 = -1.322875656

Reorder the terms:
-0.5 + z = -1.322875656

Solving
-0.5 + z = -1.322875656

Solving for variable 'z'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + z = -1.322875656 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + z = -1.322875656 + 0.5
z = -1.322875656 + 0.5

Combine like terms: -1.322875656 + 0.5 = -0.822875656
z = -0.822875656

Simplifying
z = -0.822875656

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {1.822875656, -0.822875656}
Solution
z = {1.822875656, -0.822875656}