=(3+2i)(5+3i)

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

Simplifying
0 = (3 + 2i)(5 + 3i)

Multiply (3 + 2i) * (5 + 3i)
0 = (3(5 + 3i) + 2i * (5 + 3i))
0 = ((5 * 3 + 3i * 3) + 2i * (5 + 3i))
0 = ((15 + 9i) + 2i * (5 + 3i))
0 = (15 + 9i + (5 * 2i + 3i * 2i))
0 = (15 + 9i + (10i + 6i2))

Combine like terms: 9i + 10i = 19i
0 = (15 + 19i + 6i2)

Solving
0 = 15 + 19i + 6i2

Solving for variable 'i'.

Combine like terms: 0 + -15 = -15
-15 + -19i + -6i2 = 15 + 19i + 6i2 + -15 + -19i + -6i2

Reorder the terms:
-15 + -19i + -6i2 = 15 + -15 + 19i + -19i + 6i2 + -6i2

Combine like terms: 15 + -15 = 0
-15 + -19i + -6i2 = 0 + 19i + -19i + 6i2 + -6i2
-15 + -19i + -6i2 = 19i + -19i + 6i2 + -6i2

Combine like terms: 19i + -19i = 0
-15 + -19i + -6i2 = 0 + 6i2 + -6i2
-15 + -19i + -6i2 = 6i2 + -6i2

Combine like terms: 6i2 + -6i2 = 0
-15 + -19i + -6i2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(15 + 19i + 6i2) = 0

Factor a trinomial.
-1((5 + 3i)(3 + 2i)) = 0

Ignore the factor -1.
Subproblem 1
Set the factor '(5 + 3i)' equal to zero and attempt to solve:

Simplifying
5 + 3i = 0

Solving
5 + 3i = 0

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

Add '-5' to each side of the equation.
5 + -5 + 3i = 0 + -5

Combine like terms: 5 + -5 = 0
0 + 3i = 0 + -5
3i = 0 + -5

Combine like terms: 0 + -5 = -5
3i = -5

Divide each side by '3'.
i = -1.666666667

Simplifying
i = -1.666666667
Subproblem 2
Set the factor '(3 + 2i)' equal to zero and attempt to solve:

Simplifying
3 + 2i = 0

Solving
3 + 2i = 0

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

Add '-3' to each side of the equation.
3 + -3 + 2i = 0 + -3

Combine like terms: 3 + -3 = 0
0 + 2i = 0 + -3
2i = 0 + -3

Combine like terms: 0 + -3 = -3
2i = -3

Divide each side by '2'.
i = -1.5

Simplifying
i = -1.5
Solution
i = {-1.666666667, -1.5}