3(-3-i)+(i)(1+3i)=0

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Solution for 3(-3-i)+(i)(1+3i)=0 equation:


Simplifying
3(-3 + -1i) + (i)(1 + 3i) = 0
(-3 * 3 + -1i * 3) + (i)(1 + 3i) = 0
(-9 + -3i) + (i)(1 + 3i) = 0
-9 + -3i + (1 * i + 3i * i) = 0
-9 + -3i + (1i + 3i2) = 0

Combine like terms: -3i + 1i = -2i
-9 + -2i + 3i2 = 0

Solving
-9 + -2i + 3i2 = 0

Solving for variable 'i'.

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

Divide each side by '3'.
-3 + -0.6666666667i + i2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + -0.6666666667i + 3 + i2 = 0 + 3

Reorder the terms:
-3 + 3 + -0.6666666667i + i2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + -0.6666666667i + i2 = 0 + 3
-0.6666666667i + i2 = 0 + 3

Combine like terms: 0 + 3 = 3
-0.6666666667i + i2 = 3

The i term is -0.6666666667i.  Take half its coefficient (-0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
-0.6666666667i + 0.1111111112 + i2 = 3 + 0.1111111112

Reorder the terms:
0.1111111112 + -0.6666666667i + i2 = 3 + 0.1111111112

Combine like terms: 3 + 0.1111111112 = 3.1111111112
0.1111111112 + -0.6666666667i + i2 = 3.1111111112

Factor a perfect square on the left side:
(i + -0.3333333334)(i + -0.3333333334) = 3.1111111112

Calculate the square root of the right side: 1.763834207

Break this problem into two subproblems by setting 
(i + -0.3333333334) equal to 1.763834207 and -1.763834207.

Subproblem 1

i + -0.3333333334 = 1.763834207 Simplifying i + -0.3333333334 = 1.763834207 Reorder the terms: -0.3333333334 + i = 1.763834207 Solving -0.3333333334 + i = 1.763834207 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + i = 1.763834207 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + i = 1.763834207 + 0.3333333334 i = 1.763834207 + 0.3333333334 Combine like terms: 1.763834207 + 0.3333333334 = 2.0971675404 i = 2.0971675404 Simplifying i = 2.0971675404

Subproblem 2

i + -0.3333333334 = -1.763834207 Simplifying i + -0.3333333334 = -1.763834207 Reorder the terms: -0.3333333334 + i = -1.763834207 Solving -0.3333333334 + i = -1.763834207 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '0.3333333334' to each side of the equation. -0.3333333334 + 0.3333333334 + i = -1.763834207 + 0.3333333334 Combine like terms: -0.3333333334 + 0.3333333334 = 0.0000000000 0.0000000000 + i = -1.763834207 + 0.3333333334 i = -1.763834207 + 0.3333333334 Combine like terms: -1.763834207 + 0.3333333334 = -1.4305008736 i = -1.4305008736 Simplifying i = -1.4305008736

Solution

The solution to the problem is based on the solutions from the subproblems. i = {2.0971675404, -1.4305008736}

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