9i^2+40i-16=0

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Solution for 9i^2+40i-16=0 equation: 


Simplifying
9i2 + 40i + -16 = 0

Reorder the terms:
-16 + 40i + 9i2 = 0

Solving
-16 + 40i + 9i2 = 0

Solving for variable 'i'.

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

Divide each side by '9'.
-1.777777778 + 4.444444444i + i2 = 0

Move the constant term to the right:

Add '1.777777778' to each side of the equation.
-1.777777778 + 4.444444444i + 1.777777778 + i2 = 0 + 1.777777778

Reorder the terms:
-1.777777778 + 1.777777778 + 4.444444444i + i2 = 0 + 1.777777778

Combine like terms: -1.777777778 + 1.777777778 = 0.000000000
0.000000000 + 4.444444444i + i2 = 0 + 1.777777778
4.444444444i + i2 = 0 + 1.777777778

Combine like terms: 0 + 1.777777778 = 1.777777778
4.444444444i + i2 = 1.777777778

The i term is 4.444444444i.  Take half its coefficient (2.222222222).
Square it (4.938271604) and add it to both sides.

Add '4.938271604' to each side of the equation.
4.444444444i + 4.938271604 + i2 = 1.777777778 + 4.938271604

Reorder the terms:
4.938271604 + 4.444444444i + i2 = 1.777777778 + 4.938271604

Combine like terms: 1.777777778 + 4.938271604 = 6.716049382
4.938271604 + 4.444444444i + i2 = 6.716049382

Factor a perfect square on the left side:
(i + 2.222222222)(i + 2.222222222) = 6.716049382

Calculate the square root of the right side: 2.591534175

Break this problem into two subproblems by setting 
(i + 2.222222222) equal to 2.591534175 and -2.591534175.


Subproblem 1
i + 2.222222222 = 2.591534175

Simplifying
i + 2.222222222 = 2.591534175

Reorder the terms:
2.222222222 + i = 2.591534175

Solving
2.222222222 + i = 2.591534175

Solving for variable 'i'.

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

Add '-2.222222222' to each side of the equation.
2.222222222 + -2.222222222 + i = 2.591534175 + -2.222222222

Combine like terms: 2.222222222 + -2.222222222 = 0.000000000
0.000000000 + i = 2.591534175 + -2.222222222
i = 2.591534175 + -2.222222222

Combine like terms: 2.591534175 + -2.222222222 = 0.369311953
i = 0.369311953

Simplifying
i = 0.369311953


Subproblem 2
i + 2.222222222 = -2.591534175

Simplifying
i + 2.222222222 = -2.591534175

Reorder the terms:
2.222222222 + i = -2.591534175

Solving
2.222222222 + i = -2.591534175

Solving for variable 'i'.

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

Add '-2.222222222' to each side of the equation.
2.222222222 + -2.222222222 + i = -2.591534175 + -2.222222222

Combine like terms: 2.222222222 + -2.222222222 = 0.000000000
0.000000000 + i = -2.591534175 + -2.222222222
i = -2.591534175 + -2.222222222

Combine like terms: -2.591534175 + -2.222222222 = -4.813756397
i = -4.813756397

Simplifying
i = -4.813756397


Solution
The solution to the problem is based on the solutions
from the subproblems.
i = {0.369311953, -4.813756397}

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