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

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

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

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

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

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

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

Solving
3 + -4z + z2 = 1

Solving for variable 'z'.

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

Combine like terms: 3 + -1 = 2
2 + -4z + z2 = 1 + -1

Combine like terms: 1 + -1 = 0
2 + -4z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + -4z + -2 + z2 = 0 + -2

Reorder the terms:
2 + -2 + -4z + z2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + -4z + z2 = 0 + -2
-4z + z2 = 0 + -2

Combine like terms: 0 + -2 = -2
-4z + z2 = -2

The z term is -4z.  Take half its coefficient (-2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
-4z + 4 + z2 = -2 + 4

Reorder the terms:
4 + -4z + z2 = -2 + 4

Combine like terms: -2 + 4 = 2
4 + -4z + z2 = 2

Factor a perfect square on the left side:
(z + -2)(z + -2) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(z + -2) equal to 1.414213562 and -1.414213562.

Subproblem 1
z + -2 = 1.414213562

Simplifying
z + -2 = 1.414213562

Reorder the terms:
-2 + z = 1.414213562

Solving
-2 + z = 1.414213562

Solving for variable 'z'.

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

Add '2' to each side of the equation.
-2 + 2 + z = 1.414213562 + 2

Combine like terms: -2 + 2 = 0
0 + z = 1.414213562 + 2
z = 1.414213562 + 2

Combine like terms: 1.414213562 + 2 = 3.414213562
z = 3.414213562

Simplifying
z = 3.414213562

Subproblem 2
z + -2 = -1.414213562

Simplifying
z + -2 = -1.414213562

Reorder the terms:
-2 + z = -1.414213562

Solving
-2 + z = -1.414213562

Solving for variable 'z'.

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

Add '2' to each side of the equation.
-2 + 2 + z = -1.414213562 + 2

Combine like terms: -2 + 2 = 0
0 + z = -1.414213562 + 2
z = -1.414213562 + 2

Combine like terms: -1.414213562 + 2 = 0.585786438
z = 0.585786438

Simplifying
z = 0.585786438

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {3.414213562, 0.585786438}