(3+6i)(5-i)=0

Solution for (3+6i)(5-i)=0 equation:

Simplifying
(3 + 6i)(5 + -1i) = 0

Multiply (3 + 6i) * (5 + -1i)
(3(5 + -1i) + 6i * (5 + -1i)) = 0
((5 * 3 + -1i * 3) + 6i * (5 + -1i)) = 0
((15 + -3i) + 6i * (5 + -1i)) = 0
(15 + -3i + (5 * 6i + -1i * 6i)) = 0
(15 + -3i + (30i + -6i2)) = 0

Combine like terms: -3i + 30i = 27i
(15 + 27i + -6i2) = 0

Solving
15 + 27i + -6i2 = 0

Solving for variable 'i'.

Factor out the Greatest Common Factor (GCF), '3'.
3(5 + 9i + -2i2) = 0

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

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

Simplifying
5 + -1i = 0

Solving
5 + -1i = 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 + -1i = 0 + -5

Combine like terms: 5 + -5 = 0
0 + -1i = 0 + -5
-1i = 0 + -5

Combine like terms: 0 + -5 = -5
-1i = -5

Divide each side by '-1'.
i = 5

Simplifying
i = 5
Subproblem 2
Set the factor '(1 + 2i)' equal to zero and attempt to solve:

Simplifying
1 + 2i = 0

Solving
1 + 2i = 0

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

Add '-1' to each side of the equation.
1 + -1 + 2i = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 2i = 0 + -1
2i = 0 + -1

Combine like terms: 0 + -1 = -1
2i = -1

Divide each side by '2'.
i = -0.5

Simplifying
i = -0.5
Solution
i = {5, -0.5}