(4z^3-z^2)+(z^3-3z)=

Solution for (4z^3-z^2)+(z^3-3z)= equation:

Simplifying
(4z3 + -1z2) + (z3 + -3z) = 0

Reorder the terms:
(-1z2 + 4z3) + (z3 + -3z) = 0

Remove parenthesis around (-1z2 + 4z3)
-1z2 + 4z3 + (z3 + -3z) = 0

Reorder the terms:
-1z2 + 4z3 + (-3z + z3) = 0

Remove parenthesis around (-3z + z3)
-1z2 + 4z3 + -3z + z3 = 0

Reorder the terms:
-3z + -1z2 + 4z3 + z3 = 0

Combine like terms: 4z3 + z3 = 5z3
-3z + -1z2 + 5z3 = 0

Solving
-3z + -1z2 + 5z3 = 0

Solving for variable 'z'.

Factor out the Greatest Common Factor (GCF), 'z'.
z(-3 + -1z + 5z2) = 0

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

Simplifying
z = 0

Solving
z = 0

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

Simplifying
z = 0
Subproblem 2
Set the factor '(-3 + -1z + 5z2)' equal to zero and attempt to solve:

Simplifying
-3 + -1z + 5z2 = 0

Solving
-3 + -1z + 5z2 = 0

Begin completing the square.  Divide all terms by
5 the coefficient of the squared term: 

Divide each side by '5'.
-0.6 + -0.2z + z2 = 0

Move the constant term to the right:

Add '0.6' to each side of the equation.
-0.6 + -0.2z + 0.6 + z2 = 0 + 0.6

Reorder the terms:
-0.6 + 0.6 + -0.2z + z2 = 0 + 0.6

Combine like terms: -0.6 + 0.6 = 0.0
0.0 + -0.2z + z2 = 0 + 0.6
-0.2z + z2 = 0 + 0.6

Combine like terms: 0 + 0.6 = 0.6
-0.2z + z2 = 0.6

The z term is -0.2z.  Take half its coefficient (-0.1).
Square it (0.01) and add it to both sides.

Add '0.01' to each side of the equation.
-0.2z + 0.01 + z2 = 0.6 + 0.01

Reorder the terms:
0.01 + -0.2z + z2 = 0.6 + 0.01

Combine like terms: 0.6 + 0.01 = 0.61
0.01 + -0.2z + z2 = 0.61

Factor a perfect square on the left side:
(z + -0.1)(z + -0.1) = 0.61

Calculate the square root of the right side: 0.781024968

Break this problem into two subproblems by setting 
(z + -0.1) equal to 0.781024968 and -0.781024968.

Subproblem 1
z + -0.1 = 0.781024968

Simplifying
z + -0.1 = 0.781024968

Reorder the terms:
-0.1 + z = 0.781024968

Solving
-0.1 + z = 0.781024968

Solving for variable 'z'.

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

Add '0.1' to each side of the equation.
-0.1 + 0.1 + z = 0.781024968 + 0.1

Combine like terms: -0.1 + 0.1 = 0.0
0.0 + z = 0.781024968 + 0.1
z = 0.781024968 + 0.1

Combine like terms: 0.781024968 + 0.1 = 0.881024968
z = 0.881024968

Simplifying
z = 0.881024968

Subproblem 2
z + -0.1 = -0.781024968

Simplifying
z + -0.1 = -0.781024968

Reorder the terms:
-0.1 + z = -0.781024968

Solving
-0.1 + z = -0.781024968

Solving for variable 'z'.

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

Add '0.1' to each side of the equation.
-0.1 + 0.1 + z = -0.781024968 + 0.1

Combine like terms: -0.1 + 0.1 = 0.0
0.0 + z = -0.781024968 + 0.1
z = -0.781024968 + 0.1

Combine like terms: -0.781024968 + 0.1 = -0.681024968
z = -0.681024968

Simplifying
z = -0.681024968

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {0.881024968, -0.681024968}
Solution
z = {0, 0.881024968, -0.681024968}