(3+i)(1-4i)=0

Solution for (3+i)(1-4i)=0 equation:

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

Multiply (3 + i) * (1 + -4i)
(3(1 + -4i) + i(1 + -4i)) = 0
((1 * 3 + -4i * 3) + i(1 + -4i)) = 0
((3 + -12i) + i(1 + -4i)) = 0
(3 + -12i + (1 * i + -4i * i)) = 0
(3 + -12i + (1i + -4i2)) = 0

Combine like terms: -12i + 1i = -11i
(3 + -11i + -4i2) = 0

Solving
3 + -11i + -4i2 = 0

Solving for variable 'i'.

Factor a trinomial.
(1 + -4i)(3 + i) = 0

Subproblem 1
Set the factor '(1 + -4i)' equal to zero and attempt to solve:

Simplifying
1 + -4i = 0

Solving
1 + -4i = 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 + -4i = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -4i = 0 + -1
-4i = 0 + -1

Combine like terms: 0 + -1 = -1
-4i = -1

Divide each side by '-4'.
i = 0.25

Simplifying
i = 0.25
Subproblem 2
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
Solution
i = {0.25, -3}