-42i^2-21i+36=0

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Solution for -42i^2-21i+36=0 equation: 


Simplifying
-42i2 + -21i + 36 = 0

Reorder the terms:
36 + -21i + -42i2 = 0

Solving
36 + -21i + -42i2 = 0

Solving for variable 'i'.

Factor out the Greatest Common Factor (GCF), '3'.
3(12 + -7i + -14i2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(12 + -7i + -14i2)' equal to zero and attempt to solve:

Simplifying
12 + -7i + -14i2 = 0

Solving
12 + -7i + -14i2 = 0

Begin completing the square.  Divide all terms by
-14 the coefficient of the squared term: 

Divide each side by '-14'.
-0.8571428571 + 0.5i + i2 = 0

Move the constant term to the right:

Add '0.8571428571' to each side of the equation.
-0.8571428571 + 0.5i + 0.8571428571 + i2 = 0 + 0.8571428571

Reorder the terms:
-0.8571428571 + 0.8571428571 + 0.5i + i2 = 0 + 0.8571428571

Combine like terms: -0.8571428571 + 0.8571428571 = 0.0000000000
0.0000000000 + 0.5i + i2 = 0 + 0.8571428571
0.5i + i2 = 0 + 0.8571428571

Combine like terms: 0 + 0.8571428571 = 0.8571428571
0.5i + i2 = 0.8571428571

The i term is 0.5i.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5i + 0.0625 + i2 = 0.8571428571 + 0.0625

Reorder the terms:
0.0625 + 0.5i + i2 = 0.8571428571 + 0.0625

Combine like terms: 0.8571428571 + 0.0625 = 0.9196428571
0.0625 + 0.5i + i2 = 0.9196428571

Factor a perfect square on the left side:
(i + 0.25)(i + 0.25) = 0.9196428571

Calculate the square root of the right side: 0.958980113

Break this problem into two subproblems by setting 
(i + 0.25) equal to 0.958980113 and -0.958980113.


Subproblem 1
i + 0.25 = 0.958980113

Simplifying
i + 0.25 = 0.958980113

Reorder the terms:
0.25 + i = 0.958980113

Solving
0.25 + i = 0.958980113

Solving for variable 'i'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + i = 0.958980113 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + i = 0.958980113 + -0.25
i = 0.958980113 + -0.25

Combine like terms: 0.958980113 + -0.25 = 0.708980113
i = 0.708980113

Simplifying
i = 0.708980113


Subproblem 2
i + 0.25 = -0.958980113

Simplifying
i + 0.25 = -0.958980113

Reorder the terms:
0.25 + i = -0.958980113

Solving
0.25 + i = -0.958980113

Solving for variable 'i'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + i = -0.958980113 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + i = -0.958980113 + -0.25
i = -0.958980113 + -0.25

Combine like terms: -0.958980113 + -0.25 = -1.208980113
i = -1.208980113

Simplifying
i = -1.208980113


Solution
The solution to the problem is based on the solutions
from the subproblems.
i = {0.708980113, -1.208980113}
Solution
i = {0.708980113, -1.208980113}

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