i(4-i)+3(i-3)=0

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Solution for i(4-i)+3(i-3)=0 equation:


Simplifying
i(4 + -1i) + 3(i + -3) = 0
(4 * i + -1i * i) + 3(i + -3) = 0
(4i + -1i2) + 3(i + -3) = 0

Reorder the terms:
4i + -1i2 + 3(-3 + i) = 0
4i + -1i2 + (-3 * 3 + i * 3) = 0
4i + -1i2 + (-9 + 3i) = 0

Reorder the terms:
-9 + 4i + 3i + -1i2 = 0

Combine like terms: 4i + 3i = 7i
-9 + 7i + -1i2 = 0

Solving
-9 + 7i + -1i2 = 0

Solving for variable 'i'.

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
9 + -7i + i2 = 0

Move the constant term to the right:

Add '-9' to each side of the equation.
9 + -7i + -9 + i2 = 0 + -9

Reorder the terms:
9 + -9 + -7i + i2 = 0 + -9

Combine like terms: 9 + -9 = 0
0 + -7i + i2 = 0 + -9
-7i + i2 = 0 + -9

Combine like terms: 0 + -9 = -9
-7i + i2 = -9

The i term is -7i.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7i + 12.25 + i2 = -9 + 12.25

Reorder the terms:
12.25 + -7i + i2 = -9 + 12.25

Combine like terms: -9 + 12.25 = 3.25
12.25 + -7i + i2 = 3.25

Factor a perfect square on the left side:
(i + -3.5)(i + -3.5) = 3.25

Calculate the square root of the right side: 1.802775638

Break this problem into two subproblems by setting 
(i + -3.5) equal to 1.802775638 and -1.802775638.

Subproblem 1

i + -3.5 = 1.802775638 Simplifying i + -3.5 = 1.802775638 Reorder the terms: -3.5 + i = 1.802775638 Solving -3.5 + i = 1.802775638 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + i = 1.802775638 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + i = 1.802775638 + 3.5 i = 1.802775638 + 3.5 Combine like terms: 1.802775638 + 3.5 = 5.302775638 i = 5.302775638 Simplifying i = 5.302775638

Subproblem 2

i + -3.5 = -1.802775638 Simplifying i + -3.5 = -1.802775638 Reorder the terms: -3.5 + i = -1.802775638 Solving -3.5 + i = -1.802775638 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '3.5' to each side of the equation. -3.5 + 3.5 + i = -1.802775638 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + i = -1.802775638 + 3.5 i = -1.802775638 + 3.5 Combine like terms: -1.802775638 + 3.5 = 1.697224362 i = 1.697224362 Simplifying i = 1.697224362

Solution

The solution to the problem is based on the solutions from the subproblems. i = {5.302775638, 1.697224362}

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