(z-1)(z-1)-3z=3

Solution for (z-1)(z-1)-3z=3 equation:

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

Reorder the terms:
(-1 + z)(z + -1) + -3z = 3

Reorder the terms:
(-1 + z)(-1 + z) + -3z = 3

Multiply (-1 + z) * (-1 + z)
(-1(-1 + z) + z(-1 + z)) + -3z = 3
((-1 * -1 + z * -1) + z(-1 + z)) + -3z = 3
((1 + -1z) + z(-1 + z)) + -3z = 3
(1 + -1z + (-1 * z + z * z)) + -3z = 3
(1 + -1z + (-1z + z2)) + -3z = 3

Combine like terms: -1z + -1z = -2z
(1 + -2z + z2) + -3z = 3

Reorder the terms:
1 + -2z + -3z + z2 = 3

Combine like terms: -2z + -3z = -5z
1 + -5z + z2 = 3

Solving
1 + -5z + z2 = 3

Solving for variable 'z'.

Reorder the terms:
1 + -3 + -5z + z2 = 3 + -3

Combine like terms: 1 + -3 = -2
-2 + -5z + z2 = 3 + -3

Combine like terms: 3 + -3 = 0
-2 + -5z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -5z + 2 + z2 = 0 + 2

Reorder the terms:
-2 + 2 + -5z + z2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -5z + z2 = 0 + 2
-5z + z2 = 0 + 2

Combine like terms: 0 + 2 = 2
-5z + z2 = 2

The z term is -5z.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5z + 6.25 + z2 = 2 + 6.25

Reorder the terms:
6.25 + -5z + z2 = 2 + 6.25

Combine like terms: 2 + 6.25 = 8.25
6.25 + -5z + z2 = 8.25

Factor a perfect square on the left side:
(z + -2.5)(z + -2.5) = 8.25

Calculate the square root of the right side: 2.872281323

Break this problem into two subproblems by setting 
(z + -2.5) equal to 2.872281323 and -2.872281323.

Subproblem 1
z + -2.5 = 2.872281323

Simplifying
z + -2.5 = 2.872281323

Reorder the terms:
-2.5 + z = 2.872281323

Solving
-2.5 + z = 2.872281323

Solving for variable 'z'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + z = 2.872281323 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + z = 2.872281323 + 2.5
z = 2.872281323 + 2.5

Combine like terms: 2.872281323 + 2.5 = 5.372281323
z = 5.372281323

Simplifying
z = 5.372281323

Subproblem 2
z + -2.5 = -2.872281323

Simplifying
z + -2.5 = -2.872281323

Reorder the terms:
-2.5 + z = -2.872281323

Solving
-2.5 + z = -2.872281323

Solving for variable 'z'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + z = -2.872281323 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + z = -2.872281323 + 2.5
z = -2.872281323 + 2.5

Combine like terms: -2.872281323 + 2.5 = -0.372281323
z = -0.372281323

Simplifying
z = -0.372281323

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {5.372281323, -0.372281323}