(1-j)+(1-j)(0.707-0.707j)=0

Solution for (1-j)+(1-j)(0.707-0.707j)=0 equation:

Simplifying
(1 + -1j) + (1 + -1j)(0.707 + -0.707j) = 0

Remove parenthesis around (1 + -1j)
1 + -1j + (1 + -1j)(0.707 + -0.707j) = 0

Multiply (1 + -1j) * (0.707 + -0.707j)
1 + -1j + (1(0.707 + -0.707j) + -1j * (0.707 + -0.707j)) = 0
1 + -1j + ((0.707 * 1 + -0.707j * 1) + -1j * (0.707 + -0.707j)) = 0
1 + -1j + ((0.707 + -0.707j) + -1j * (0.707 + -0.707j)) = 0
1 + -1j + (0.707 + -0.707j + (0.707 * -1j + -0.707j * -1j)) = 0
1 + -1j + (0.707 + -0.707j + (-0.707j + 0.707j2)) = 0

Combine like terms: -0.707j + -0.707j = -1.414j
1 + -1j + (0.707 + -1.414j + 0.707j2) = 0

Reorder the terms:
1 + 0.707 + -1j + -1.414j + 0.707j2 = 0

Combine like terms: 1 + 0.707 = 1.707
1.707 + -1j + -1.414j + 0.707j2 = 0

Combine like terms: -1j + -1.414j = -2.414j
1.707 + -2.414j + 0.707j2 = 0

Solving
1.707 + -2.414j + 0.707j2 = 0

Solving for variable 'j'.

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

Divide each side by '0.707'.
2.414427157 + -3.414427157j + j2 = 0

Move the constant term to the right:

Add '-2.414427157' to each side of the equation.
2.414427157 + -3.414427157j + -2.414427157 + j2 = 0 + -2.414427157

Reorder the terms:
2.414427157 + -2.414427157 + -3.414427157j + j2 = 0 + -2.414427157

Combine like terms: 2.414427157 + -2.414427157 = 0.000000000
0.000000000 + -3.414427157j + j2 = 0 + -2.414427157
-3.414427157j + j2 = 0 + -2.414427157

Combine like terms: 0 + -2.414427157 = -2.414427157
-3.414427157j + j2 = -2.414427157

The j term is -3.414427157j.  Take half its coefficient (-1.707213579).
Square it (2.914578204) and add it to both sides.

Add '2.914578204' to each side of the equation.
-3.414427157j + 2.914578204 + j2 = -2.414427157 + 2.914578204

Reorder the terms:
2.914578204 + -3.414427157j + j2 = -2.414427157 + 2.914578204

Combine like terms: -2.414427157 + 2.914578204 = 0.500151047
2.914578204 + -3.414427157j + j2 = 0.500151047

Factor a perfect square on the left side:
(j + -1.707213579)(j + -1.707213579) = 0.500151047

Calculate the square root of the right side: 0.707213579

Break this problem into two subproblems by setting 
(j + -1.707213579) equal to 0.707213579 and -0.707213579.

Subproblem 1
j + -1.707213579 = 0.707213579

Simplifying
j + -1.707213579 = 0.707213579

Reorder the terms:
-1.707213579 + j = 0.707213579

Solving
-1.707213579 + j = 0.707213579

Solving for variable 'j'.

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

Add '1.707213579' to each side of the equation.
-1.707213579 + 1.707213579 + j = 0.707213579 + 1.707213579

Combine like terms: -1.707213579 + 1.707213579 = 0.000000000
0.000000000 + j = 0.707213579 + 1.707213579
j = 0.707213579 + 1.707213579

Combine like terms: 0.707213579 + 1.707213579 = 2.414427158
j = 2.414427158

Simplifying
j = 2.414427158

Subproblem 2
j + -1.707213579 = -0.707213579

Simplifying
j + -1.707213579 = -0.707213579

Reorder the terms:
-1.707213579 + j = -0.707213579

Solving
-1.707213579 + j = -0.707213579

Solving for variable 'j'.

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

Add '1.707213579' to each side of the equation.
-1.707213579 + 1.707213579 + j = -0.707213579 + 1.707213579

Combine like terms: -1.707213579 + 1.707213579 = 0.000000000
0.000000000 + j = -0.707213579 + 1.707213579
j = -0.707213579 + 1.707213579

Combine like terms: -0.707213579 + 1.707213579 = 1
j = 1

Simplifying
j = 1

Solution
The solution to the problem is based on the solutions
from the subproblems.
j = {2.414427158, 1}