(3+i)(6+4i)=0

Solution for (3+i)(6+4i)=0 equation:

Simplifying
(3 + i)(6 + 4i) = 0

Multiply (3 + i) * (6 + 4i)
(3(6 + 4i) + i(6 + 4i)) = 0
((6 * 3 + 4i * 3) + i(6 + 4i)) = 0
((18 + 12i) + i(6 + 4i)) = 0
(18 + 12i + (6 * i + 4i * i)) = 0
(18 + 12i + (6i + 4i2)) = 0

Combine like terms: 12i + 6i = 18i
(18 + 18i + 4i2) = 0

Solving
18 + 18i + 4i2 = 0

Solving for variable 'i'.

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

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

Ignore the factor 2.
Subproblem 1
Set the factor '(3 + i)' equal to zero and attempt to solve:

Simplifying
3 + i = 0

Solving
3 + i = 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 + i = 0 + -3

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

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

Simplifying
i = -3
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 = {-3, -1.5}