3z-8=7z(z-3)

Solution for 3z-8=7z(z-3) equation:

Simplifying
3z + -8 = 7z(z + -3)

Reorder the terms:
-8 + 3z = 7z(z + -3)

Reorder the terms:
-8 + 3z = 7z(-3 + z)
-8 + 3z = (-3 * 7z + z * 7z)
-8 + 3z = (-21z + 7z2)

Solving
-8 + 3z = -21z + 7z2

Solving for variable 'z'.

Combine like terms: 3z + 21z = 24z
-8 + 24z + -7z2 = -21z + 7z2 + 21z + -7z2

Reorder the terms:
-8 + 24z + -7z2 = -21z + 21z + 7z2 + -7z2

Combine like terms: -21z + 21z = 0
-8 + 24z + -7z2 = 0 + 7z2 + -7z2
-8 + 24z + -7z2 = 7z2 + -7z2

Combine like terms: 7z2 + -7z2 = 0
-8 + 24z + -7z2 = 0

Begin completing the square.  Divide all terms by
-7 the coefficient of the squared term: 

Divide each side by '-7'.
1.142857143 + -3.428571429z + z2 = 0

Move the constant term to the right:

Add '-1.142857143' to each side of the equation.
1.142857143 + -3.428571429z + -1.142857143 + z2 = 0 + -1.142857143

Reorder the terms:
1.142857143 + -1.142857143 + -3.428571429z + z2 = 0 + -1.142857143

Combine like terms: 1.142857143 + -1.142857143 = 0.000000000
0.000000000 + -3.428571429z + z2 = 0 + -1.142857143
-3.428571429z + z2 = 0 + -1.142857143

Combine like terms: 0 + -1.142857143 = -1.142857143
-3.428571429z + z2 = -1.142857143

The z term is -3.428571429z.  Take half its coefficient (-1.714285715).
Square it (2.938775513) and add it to both sides.

Add '2.938775513' to each side of the equation.
-3.428571429z + 2.938775513 + z2 = -1.142857143 + 2.938775513

Reorder the terms:
2.938775513 + -3.428571429z + z2 = -1.142857143 + 2.938775513

Combine like terms: -1.142857143 + 2.938775513 = 1.79591837
2.938775513 + -3.428571429z + z2 = 1.79591837

Factor a perfect square on the left side:
(z + -1.714285715)(z + -1.714285715) = 1.79591837

Calculate the square root of the right side: 1.34011879

Break this problem into two subproblems by setting 
(z + -1.714285715) equal to 1.34011879 and -1.34011879.

Subproblem 1
z + -1.714285715 = 1.34011879

Simplifying
z + -1.714285715 = 1.34011879

Reorder the terms:
-1.714285715 + z = 1.34011879

Solving
-1.714285715 + z = 1.34011879

Solving for variable 'z'.

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

Add '1.714285715' to each side of the equation.
-1.714285715 + 1.714285715 + z = 1.34011879 + 1.714285715

Combine like terms: -1.714285715 + 1.714285715 = 0.000000000
0.000000000 + z = 1.34011879 + 1.714285715
z = 1.34011879 + 1.714285715

Combine like terms: 1.34011879 + 1.714285715 = 3.054404505
z = 3.054404505

Simplifying
z = 3.054404505

Subproblem 2
z + -1.714285715 = -1.34011879

Simplifying
z + -1.714285715 = -1.34011879

Reorder the terms:
-1.714285715 + z = -1.34011879

Solving
-1.714285715 + z = -1.34011879

Solving for variable 'z'.

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

Add '1.714285715' to each side of the equation.
-1.714285715 + 1.714285715 + z = -1.34011879 + 1.714285715

Combine like terms: -1.714285715 + 1.714285715 = 0.000000000
0.000000000 + z = -1.34011879 + 1.714285715
z = -1.34011879 + 1.714285715

Combine like terms: -1.34011879 + 1.714285715 = 0.374166925
z = 0.374166925

Simplifying
z = 0.374166925

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {3.054404505, 0.374166925}