0=9+3i(7-2i)

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Solution for 0=9+3i(7-2i) equation: 


Simplifying
0 = 9 + 3i(7 + -2i)
0 = 9 + (7 * 3i + -2i * 3i)
0 = 9 + (21i + -6i2)

Solving
0 = 9 + 21i + -6i2

Solving for variable 'i'.

Combine like terms: 0 + -9 = -9
-9 + -21i + 6i2 = 9 + 21i + -6i2 + -9 + -21i + 6i2

Reorder the terms:
-9 + -21i + 6i2 = 9 + -9 + 21i + -21i + -6i2 + 6i2

Combine like terms: 9 + -9 = 0
-9 + -21i + 6i2 = 0 + 21i + -21i + -6i2 + 6i2
-9 + -21i + 6i2 = 21i + -21i + -6i2 + 6i2

Combine like terms: 21i + -21i = 0
-9 + -21i + 6i2 = 0 + -6i2 + 6i2
-9 + -21i + 6i2 = -6i2 + 6i2

Combine like terms: -6i2 + 6i2 = 0
-9 + -21i + 6i2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-3 + -7i + 2i2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(-3 + -7i + 2i2)' equal to zero and attempt to solve:

Simplifying
-3 + -7i + 2i2 = 0

Solving
-3 + -7i + 2i2 = 0

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

Divide each side by '2'.
-1.5 + -3.5i + i2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + -3.5i + 1.5 + i2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + -3.5i + i2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + -3.5i + i2 = 0 + 1.5
-3.5i + i2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
-3.5i + i2 = 1.5

The i term is -3.5i.  Take half its coefficient (-1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
-3.5i + 3.0625 + i2 = 1.5 + 3.0625

Reorder the terms:
3.0625 + -3.5i + i2 = 1.5 + 3.0625

Combine like terms: 1.5 + 3.0625 = 4.5625
3.0625 + -3.5i + i2 = 4.5625

Factor a perfect square on the left side:
(i + -1.75)(i + -1.75) = 4.5625

Calculate the square root of the right side: 2.136000936

Break this problem into two subproblems by setting 
(i + -1.75) equal to 2.136000936 and -2.136000936.


Subproblem 1
i + -1.75 = 2.136000936

Simplifying
i + -1.75 = 2.136000936

Reorder the terms:
-1.75 + i = 2.136000936

Solving
-1.75 + i = 2.136000936

Solving for variable 'i'.

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

Add '1.75' to each side of the equation.
-1.75 + 1.75 + i = 2.136000936 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + i = 2.136000936 + 1.75
i = 2.136000936 + 1.75

Combine like terms: 2.136000936 + 1.75 = 3.886000936
i = 3.886000936

Simplifying
i = 3.886000936


Subproblem 2
i + -1.75 = -2.136000936

Simplifying
i + -1.75 = -2.136000936

Reorder the terms:
-1.75 + i = -2.136000936

Solving
-1.75 + i = -2.136000936

Solving for variable 'i'.

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

Add '1.75' to each side of the equation.
-1.75 + 1.75 + i = -2.136000936 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + i = -2.136000936 + 1.75
i = -2.136000936 + 1.75

Combine like terms: -2.136000936 + 1.75 = -0.386000936
i = -0.386000936

Simplifying
i = -0.386000936


Solution
The solution to the problem is based on the solutions
from the subproblems.
i = {3.886000936, -0.386000936}
Solution
i = {3.886000936, -0.386000936}

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