z^2(-1-4i)z(5+5i)=0

Solution for z^2(-1-4i)z(5+5i)=0 equation:

Simplifying
z2(-1 + -4i) * z(5 + 5i) = 0

Reorder the terms for easier multiplication:
z2 * z(-1 + -4i)(5 + 5i) = 0

Multiply z2 * z
z3(-1 + -4i)(5 + 5i) = 0

Multiply (-1 + -4i) * (5 + 5i)
z3(-1(5 + 5i) + -4i * (5 + 5i)) = 0
z3((5 * -1 + 5i * -1) + -4i * (5 + 5i)) = 0
z3((-5 + -5i) + -4i * (5 + 5i)) = 0
z3(-5 + -5i + (5 * -4i + 5i * -4i)) = 0
z3(-5 + -5i + (-20i + -20i2)) = 0

Combine like terms: -5i + -20i = -25i
z3(-5 + -25i + -20i2) = 0
(-5 * z3 + -25i * z3 + -20i2 * z3) = 0

Reorder the terms:
(-25iz3 + -20i2z3 + -5z3) = 0
(-25iz3 + -20i2z3 + -5z3) = 0

Solving
-25iz3 + -20i2z3 + -5z3 = 0

Solving for variable 'i'.

Factor out the Greatest Common Factor (GCF), '-5z3'.
-5z3(5i + 4i2 + 1) = 0

Factor a trinomial.
-5z3((4i + 1)(i + 1)) = 0

Ignore the factor -5.
Subproblem 1
Set the factor 'z3' equal to zero and attempt to solve:

Simplifying
z3 = 0

Solving
z3 = 0

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

Add '-1z3' to each side of the equation.
z3 + -1z3 = 0 + -1z3
Remove the zero:
0 = -1z3

Simplifying
0 = -1z3

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.
Subproblem 2
Set the factor '(4i + 1)' equal to zero and attempt to solve:

Simplifying
4i + 1 = 0

Reorder the terms:
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 3
Set the factor '(i + 1)' equal to zero and attempt to solve:

Simplifying
i + 1 = 0

Reorder the terms:
1 + i = 0

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

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

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

Simplifying
i = -1
Solution
i = {-0.25, -1}