(3+4i)-(7-5i)+2i(9+12i)=0

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Solution for (3+4i)-(7-5i)+2i(9+12i)=0 equation:


Simplifying
(3 + 4i) + -1(7 + -5i) + 2i(9 + 12i) = 0

Remove parenthesis around (3 + 4i)
3 + 4i + -1(7 + -5i) + 2i(9 + 12i) = 0
3 + 4i + (7 * -1 + -5i * -1) + 2i(9 + 12i) = 0
3 + 4i + (-7 + 5i) + 2i(9 + 12i) = 0
3 + 4i + -7 + 5i + (9 * 2i + 12i * 2i) = 0
3 + 4i + -7 + 5i + (18i + 24i2) = 0

Reorder the terms:
3 + -7 + 4i + 5i + 18i + 24i2 = 0

Combine like terms: 3 + -7 = -4
-4 + 4i + 5i + 18i + 24i2 = 0

Combine like terms: 4i + 5i = 9i
-4 + 9i + 18i + 24i2 = 0

Combine like terms: 9i + 18i = 27i
-4 + 27i + 24i2 = 0

Solving
-4 + 27i + 24i2 = 0

Solving for variable 'i'.

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

Divide each side by '24'.
-0.1666666667 + 1.125i + i2 = 0

Move the constant term to the right:

Add '0.1666666667' to each side of the equation.
-0.1666666667 + 1.125i + 0.1666666667 + i2 = 0 + 0.1666666667

Reorder the terms:
-0.1666666667 + 0.1666666667 + 1.125i + i2 = 0 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + 1.125i + i2 = 0 + 0.1666666667
1.125i + i2 = 0 + 0.1666666667

Combine like terms: 0 + 0.1666666667 = 0.1666666667
1.125i + i2 = 0.1666666667

The i term is 1.125i.  Take half its coefficient (0.5625).
Square it (0.31640625) and add it to both sides.

Add '0.31640625' to each side of the equation.
1.125i + 0.31640625 + i2 = 0.1666666667 + 0.31640625

Reorder the terms:
0.31640625 + 1.125i + i2 = 0.1666666667 + 0.31640625

Combine like terms: 0.1666666667 + 0.31640625 = 0.4830729167
0.31640625 + 1.125i + i2 = 0.4830729167

Factor a perfect square on the left side:
(i + 0.5625)(i + 0.5625) = 0.4830729167

Calculate the square root of the right side: 0.695034472

Break this problem into two subproblems by setting 
(i + 0.5625) equal to 0.695034472 and -0.695034472.

Subproblem 1

i + 0.5625 = 0.695034472 Simplifying i + 0.5625 = 0.695034472 Reorder the terms: 0.5625 + i = 0.695034472 Solving 0.5625 + i = 0.695034472 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-0.5625' to each side of the equation. 0.5625 + -0.5625 + i = 0.695034472 + -0.5625 Combine like terms: 0.5625 + -0.5625 = 0.0000 0.0000 + i = 0.695034472 + -0.5625 i = 0.695034472 + -0.5625 Combine like terms: 0.695034472 + -0.5625 = 0.132534472 i = 0.132534472 Simplifying i = 0.132534472

Subproblem 2

i + 0.5625 = -0.695034472 Simplifying i + 0.5625 = -0.695034472 Reorder the terms: 0.5625 + i = -0.695034472 Solving 0.5625 + i = -0.695034472 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-0.5625' to each side of the equation. 0.5625 + -0.5625 + i = -0.695034472 + -0.5625 Combine like terms: 0.5625 + -0.5625 = 0.0000 0.0000 + i = -0.695034472 + -0.5625 i = -0.695034472 + -0.5625 Combine like terms: -0.695034472 + -0.5625 = -1.257534472 i = -1.257534472 Simplifying i = -1.257534472

Solution

The solution to the problem is based on the solutions from the subproblems. i = {0.132534472, -1.257534472}

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