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

Solution for (5+i)(1-4i)=0 equation:

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

Multiply (5 + i) * (1 + -4i)
(5(1 + -4i) + i(1 + -4i)) = 0
((1 * 5 + -4i * 5) + i(1 + -4i)) = 0
((5 + -20i) + i(1 + -4i)) = 0
(5 + -20i + (1 * i + -4i * i)) = 0
(5 + -20i + (1i + -4i2)) = 0

Combine like terms: -20i + 1i = -19i
(5 + -19i + -4i2) = 0

Solving
5 + -19i + -4i2 = 0

Solving for variable 'i'.

Factor a trinomial.
(1 + -4i)(5 + 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 '(5 + i)' equal to zero and attempt to solve:

Simplifying
5 + i = 0

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

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

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

Simplifying
i = -5
Solution
i = {0.25, -5}