5z-3=z(z-3)

Solution for 5z-3=z(z-3) equation:

Simplifying
5z + -3 = z(z + -3)

Reorder the terms:
-3 + 5z = z(z + -3)

Reorder the terms:
-3 + 5z = z(-3 + z)
-3 + 5z = (-3 * z + z * z)
-3 + 5z = (-3z + z2)

Solving
-3 + 5z = -3z + z2

Solving for variable 'z'.

Combine like terms: 5z + 3z = 8z
-3 + 8z + -1z2 = -3z + z2 + 3z + -1z2

Reorder the terms:
-3 + 8z + -1z2 = -3z + 3z + z2 + -1z2

Combine like terms: -3z + 3z = 0
-3 + 8z + -1z2 = 0 + z2 + -1z2
-3 + 8z + -1z2 = z2 + -1z2

Combine like terms: z2 + -1z2 = 0
-3 + 8z + -1z2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
3 + -8z + z2 = 0

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + -8z + -3 + z2 = 0 + -3

Reorder the terms:
3 + -3 + -8z + z2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -8z + z2 = 0 + -3
-8z + z2 = 0 + -3

Combine like terms: 0 + -3 = -3
-8z + z2 = -3

The z term is -8z.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8z + 16 + z2 = -3 + 16

Reorder the terms:
16 + -8z + z2 = -3 + 16

Combine like terms: -3 + 16 = 13
16 + -8z + z2 = 13

Factor a perfect square on the left side:
(z + -4)(z + -4) = 13

Calculate the square root of the right side: 3.605551275

Break this problem into two subproblems by setting 
(z + -4) equal to 3.605551275 and -3.605551275.

Subproblem 1
z + -4 = 3.605551275

Simplifying
z + -4 = 3.605551275

Reorder the terms:
-4 + z = 3.605551275

Solving
-4 + z = 3.605551275

Solving for variable 'z'.

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

Add '4' to each side of the equation.
-4 + 4 + z = 3.605551275 + 4

Combine like terms: -4 + 4 = 0
0 + z = 3.605551275 + 4
z = 3.605551275 + 4

Combine like terms: 3.605551275 + 4 = 7.605551275
z = 7.605551275

Simplifying
z = 7.605551275

Subproblem 2
z + -4 = -3.605551275

Simplifying
z + -4 = -3.605551275

Reorder the terms:
-4 + z = -3.605551275

Solving
-4 + z = -3.605551275

Solving for variable 'z'.

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

Add '4' to each side of the equation.
-4 + 4 + z = -3.605551275 + 4

Combine like terms: -4 + 4 = 0
0 + z = -3.605551275 + 4
z = -3.605551275 + 4

Combine like terms: -3.605551275 + 4 = 0.394448725
z = 0.394448725

Simplifying
z = 0.394448725

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {7.605551275, 0.394448725}