(6+3j)*(7+j)=1

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Solution for (6+3j)*(7+j)=1 equation: 


Simplifying
(6 + 3j)(7 + j) = 1

Multiply (6 + 3j) * (7 + j)
(6(7 + j) + 3j * (7 + j)) = 1
((7 * 6 + j * 6) + 3j * (7 + j)) = 1
((42 + 6j) + 3j * (7 + j)) = 1
(42 + 6j + (7 * 3j + j * 3j)) = 1
(42 + 6j + (21j + 3j2)) = 1

Combine like terms: 6j + 21j = 27j
(42 + 27j + 3j2) = 1

Solving
42 + 27j + 3j2 = 1

Solving for variable 'j'.

Reorder the terms:
42 + -1 + 27j + 3j2 = 1 + -1

Combine like terms: 42 + -1 = 41
41 + 27j + 3j2 = 1 + -1

Combine like terms: 1 + -1 = 0
41 + 27j + 3j2 = 0

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

Divide each side by '3'.
13.66666667 + 9j + j2 = 0

Move the constant term to the right:

Add '-13.66666667' to each side of the equation.
13.66666667 + 9j + -13.66666667 + j2 = 0 + -13.66666667

Reorder the terms:
13.66666667 + -13.66666667 + 9j + j2 = 0 + -13.66666667

Combine like terms: 13.66666667 + -13.66666667 = 0.00000000
0.00000000 + 9j + j2 = 0 + -13.66666667
9j + j2 = 0 + -13.66666667

Combine like terms: 0 + -13.66666667 = -13.66666667
9j + j2 = -13.66666667

The j term is 9j.  Take half its coefficient (4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
9j + 20.25 + j2 = -13.66666667 + 20.25

Reorder the terms:
20.25 + 9j + j2 = -13.66666667 + 20.25

Combine like terms: -13.66666667 + 20.25 = 6.58333333
20.25 + 9j + j2 = 6.58333333

Factor a perfect square on the left side:
(j + 4.5)(j + 4.5) = 6.58333333

Calculate the square root of the right side: 2.565800719

Break this problem into two subproblems by setting 
(j + 4.5) equal to 2.565800719 and -2.565800719.


Subproblem 1
j + 4.5 = 2.565800719

Simplifying
j + 4.5 = 2.565800719

Reorder the terms:
4.5 + j = 2.565800719

Solving
4.5 + j = 2.565800719

Solving for variable 'j'.

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

Add '-4.5' to each side of the equation.
4.5 + -4.5 + j = 2.565800719 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + j = 2.565800719 + -4.5
j = 2.565800719 + -4.5

Combine like terms: 2.565800719 + -4.5 = -1.934199281
j = -1.934199281

Simplifying
j = -1.934199281


Subproblem 2
j + 4.5 = -2.565800719

Simplifying
j + 4.5 = -2.565800719

Reorder the terms:
4.5 + j = -2.565800719

Solving
4.5 + j = -2.565800719

Solving for variable 'j'.

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

Add '-4.5' to each side of the equation.
4.5 + -4.5 + j = -2.565800719 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + j = -2.565800719 + -4.5
j = -2.565800719 + -4.5

Combine like terms: -2.565800719 + -4.5 = -7.065800719
j = -7.065800719

Simplifying
j = -7.065800719


Solution
The solution to the problem is based on the solutions
from the subproblems.
j = {-1.934199281, -7.065800719}

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