5z(z-1)=4(z+4)

Solution for 5z(z-1)=4(z+4) equation:

Simplifying
5z(z + -1) = 4(z + 4)

Reorder the terms:
5z(-1 + z) = 4(z + 4)
(-1 * 5z + z * 5z) = 4(z + 4)
(-5z + 5z2) = 4(z + 4)

Reorder the terms:
-5z + 5z2 = 4(4 + z)
-5z + 5z2 = (4 * 4 + z * 4)
-5z + 5z2 = (16 + 4z)

Solving
-5z + 5z2 = 16 + 4z

Solving for variable 'z'.

Reorder the terms:
-16 + -5z + -4z + 5z2 = 16 + 4z + -16 + -4z

Combine like terms: -5z + -4z = -9z
-16 + -9z + 5z2 = 16 + 4z + -16 + -4z

Reorder the terms:
-16 + -9z + 5z2 = 16 + -16 + 4z + -4z

Combine like terms: 16 + -16 = 0
-16 + -9z + 5z2 = 0 + 4z + -4z
-16 + -9z + 5z2 = 4z + -4z

Combine like terms: 4z + -4z = 0
-16 + -9z + 5z2 = 0

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

Divide each side by '5'.
-3.2 + -1.8z + z2 = 0

Move the constant term to the right:

Add '3.2' to each side of the equation.
-3.2 + -1.8z + 3.2 + z2 = 0 + 3.2

Reorder the terms:
-3.2 + 3.2 + -1.8z + z2 = 0 + 3.2

Combine like terms: -3.2 + 3.2 = 0.0
0.0 + -1.8z + z2 = 0 + 3.2
-1.8z + z2 = 0 + 3.2

Combine like terms: 0 + 3.2 = 3.2
-1.8z + z2 = 3.2

The z term is -1.8z.  Take half its coefficient (-0.9).
Square it (0.81) and add it to both sides.

Add '0.81' to each side of the equation.
-1.8z + 0.81 + z2 = 3.2 + 0.81

Reorder the terms:
0.81 + -1.8z + z2 = 3.2 + 0.81

Combine like terms: 3.2 + 0.81 = 4.01
0.81 + -1.8z + z2 = 4.01

Factor a perfect square on the left side:
(z + -0.9)(z + -0.9) = 4.01

Calculate the square root of the right side: 2.002498439

Break this problem into two subproblems by setting 
(z + -0.9) equal to 2.002498439 and -2.002498439.

Subproblem 1
z + -0.9 = 2.002498439

Simplifying
z + -0.9 = 2.002498439

Reorder the terms:
-0.9 + z = 2.002498439

Solving
-0.9 + z = 2.002498439

Solving for variable 'z'.

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

Add '0.9' to each side of the equation.
-0.9 + 0.9 + z = 2.002498439 + 0.9

Combine like terms: -0.9 + 0.9 = 0.0
0.0 + z = 2.002498439 + 0.9
z = 2.002498439 + 0.9

Combine like terms: 2.002498439 + 0.9 = 2.902498439
z = 2.902498439

Simplifying
z = 2.902498439

Subproblem 2
z + -0.9 = -2.002498439

Simplifying
z + -0.9 = -2.002498439

Reorder the terms:
-0.9 + z = -2.002498439

Solving
-0.9 + z = -2.002498439

Solving for variable 'z'.

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

Add '0.9' to each side of the equation.
-0.9 + 0.9 + z = -2.002498439 + 0.9

Combine like terms: -0.9 + 0.9 = 0.0
0.0 + z = -2.002498439 + 0.9
z = -2.002498439 + 0.9

Combine like terms: -2.002498439 + 0.9 = -1.102498439
z = -1.102498439

Simplifying
z = -1.102498439

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {2.902498439, -1.102498439}