z^4+4z^3=5z^2

Solution for z^4+4z^3=5z^2 equation:

Simplifying
z4 + 4z3 = 5z2

Reorder the terms:
4z3 + z4 = 5z2

Solving
4z3 + z4 = 5z2

Solving for variable 'z'.

Reorder the terms:
-5z2 + 4z3 + z4 = 5z2 + -5z2

Combine like terms: 5z2 + -5z2 = 0
-5z2 + 4z3 + z4 = 0

Factor out the Greatest Common Factor (GCF), 'z2'.
z2(-5 + 4z + z2) = 0

Factor a trinomial.
z2((-5 + -1z)(1 + -1z)) = 0

Subproblem 1
Set the factor 'z2' equal to zero and attempt to solve:

Simplifying
z2 = 0

Solving
z2 = 0

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

Simplifying
z2 = 0

Take the square root of each side:
z = {0}
Subproblem 2
Set the factor '(-5 + -1z)' equal to zero and attempt to solve:

Simplifying
-5 + -1z = 0

Solving
-5 + -1z = 0

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

Add '5' to each side of the equation.
-5 + 5 + -1z = 0 + 5

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

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

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

Simplifying
z = -5
Subproblem 3
Set the factor '(1 + -1z)' equal to zero and attempt to solve:

Simplifying
1 + -1z = 0

Solving
1 + -1z = 0

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

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

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

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

Divide each side by '-1'.
z = 1

Simplifying
z = 1
Solution
z = {0, -5, 1}