(5+i)*(3-2i)=4

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Solution for (5+i)*(3-2i)=4 equation:


Simplifying
(5 + i)(3 + -2i) = 4

Multiply (5 + i) * (3 + -2i)
(5(3 + -2i) + i(3 + -2i)) = 4
((3 * 5 + -2i * 5) + i(3 + -2i)) = 4
((15 + -10i) + i(3 + -2i)) = 4
(15 + -10i + (3 * i + -2i * i)) = 4
(15 + -10i + (3i + -2i2)) = 4

Combine like terms: -10i + 3i = -7i
(15 + -7i + -2i2) = 4

Solving
15 + -7i + -2i2 = 4

Solving for variable 'i'.

Reorder the terms:
15 + -4 + -7i + -2i2 = 4 + -4

Combine like terms: 15 + -4 = 11
11 + -7i + -2i2 = 4 + -4

Combine like terms: 4 + -4 = 0
11 + -7i + -2i2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
-5.5 + 3.5i + i2 = 0

Move the constant term to the right:

Add '5.5' to each side of the equation.
-5.5 + 3.5i + 5.5 + i2 = 0 + 5.5

Reorder the terms:
-5.5 + 5.5 + 3.5i + i2 = 0 + 5.5

Combine like terms: -5.5 + 5.5 = 0.0
0.0 + 3.5i + i2 = 0 + 5.5
3.5i + i2 = 0 + 5.5

Combine like terms: 0 + 5.5 = 5.5
3.5i + i2 = 5.5

The i term is 3.5i.  Take half its coefficient (1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
3.5i + 3.0625 + i2 = 5.5 + 3.0625

Reorder the terms:
3.0625 + 3.5i + i2 = 5.5 + 3.0625

Combine like terms: 5.5 + 3.0625 = 8.5625
3.0625 + 3.5i + i2 = 8.5625

Factor a perfect square on the left side:
(i + 1.75)(i + 1.75) = 8.5625

Calculate the square root of the right side: 2.926174978

Break this problem into two subproblems by setting 
(i + 1.75) equal to 2.926174978 and -2.926174978.

Subproblem 1

i + 1.75 = 2.926174978 Simplifying i + 1.75 = 2.926174978 Reorder the terms: 1.75 + i = 2.926174978 Solving 1.75 + i = 2.926174978 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + i = 2.926174978 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + i = 2.926174978 + -1.75 i = 2.926174978 + -1.75 Combine like terms: 2.926174978 + -1.75 = 1.176174978 i = 1.176174978 Simplifying i = 1.176174978

Subproblem 2

i + 1.75 = -2.926174978 Simplifying i + 1.75 = -2.926174978 Reorder the terms: 1.75 + i = -2.926174978 Solving 1.75 + i = -2.926174978 Solving for variable 'i'. Move all terms containing i to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + i = -2.926174978 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + i = -2.926174978 + -1.75 i = -2.926174978 + -1.75 Combine like terms: -2.926174978 + -1.75 = -4.676174978 i = -4.676174978 Simplifying i = -4.676174978

Solution

The solution to the problem is based on the solutions from the subproblems. i = {1.176174978, -4.676174978}

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