(-3i)(6i)-3(-7+6i)=0

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Solution for (-3i)(6i)-3(-7+6i)=0 equation: 


Simplifying
(-3i)(6i) + -3(-7 + 6i) = 0

Remove parenthesis around (-3i)
-3i(6i) + -3(-7 + 6i) = 0

Remove parenthesis around (6i)
-3i * 6i + -3(-7 + 6i) = 0

Reorder the terms for easier multiplication:
-3 * 6i * i + -3(-7 + 6i) = 0

Multiply -3 * 6
-18i * i + -3(-7 + 6i) = 0

Multiply i * i
-18i2 + -3(-7 + 6i) = 0
-18i2 + (-7 * -3 + 6i * -3) = 0
-18i2 + (21 + -18i) = 0

Reorder the terms:
21 + -18i + -18i2 = 0

Solving
21 + -18i + -18i2 = 0

Solving for variable 'i'.

Factor out the Greatest Common Factor (GCF), '3'.
3(7 + -6i + -6i2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(7 + -6i + -6i2)' equal to zero and attempt to solve:

Simplifying
7 + -6i + -6i2 = 0

Solving
7 + -6i + -6i2 = 0

Begin completing the square.  Divide all terms by
-6 the coefficient of the squared term: 

Divide each side by '-6'.
-1.166666667 + i + i2 = 0

Move the constant term to the right:

Add '1.166666667' to each side of the equation.
-1.166666667 + i + 1.166666667 + i2 = 0 + 1.166666667

Reorder the terms:
-1.166666667 + 1.166666667 + i + i2 = 0 + 1.166666667

Combine like terms: -1.166666667 + 1.166666667 = 0.000000000
0.000000000 + i + i2 = 0 + 1.166666667
i + i2 = 0 + 1.166666667

Combine like terms: 0 + 1.166666667 = 1.166666667
i + i2 = 1.166666667

The i term is i.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
 + 0.25 + i2 = 1.166666667 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + i2 = 1.166666667 + 0.25

Combine like terms: 1.166666667 + 0.25 = 1.416666667
1.25 + i2 = 1.416666667

Factor a perfect square on the left side:
(i + 0.5)(i + 0.5) = 1.416666667

Calculate the square root of the right side: 1.190238072

Break this problem into two subproblems by setting 
(i + 0.5) equal to 1.190238072 and -1.190238072.


Subproblem 1
i + 0.5 = 1.190238072

Simplifying
i + 0.5 = 1.190238072

Reorder the terms:
0.5 + i = 1.190238072

Solving
0.5 + i = 1.190238072

Solving for variable 'i'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + i = 1.190238072 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + i = 1.190238072 + -0.5
i = 1.190238072 + -0.5

Combine like terms: 1.190238072 + -0.5 = 0.690238072
i = 0.690238072

Simplifying
i = 0.690238072


Subproblem 2
i + 0.5 = -1.190238072

Simplifying
i + 0.5 = -1.190238072

Reorder the terms:
0.5 + i = -1.190238072

Solving
0.5 + i = -1.190238072

Solving for variable 'i'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + i = -1.190238072 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + i = -1.190238072 + -0.5
i = -1.190238072 + -0.5

Combine like terms: -1.190238072 + -0.5 = -1.690238072
i = -1.690238072

Simplifying
i = -1.690238072


Solution
The solution to the problem is based on the solutions
from the subproblems.
i = {0.690238072, -1.690238072}
Solution
i = {0.690238072, -1.690238072}

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