z(z-3)(2z+1)=0

Solution for z(z-3)(2z+1)=0 equation:

Simplifying
z(z + -3)(2z + 1) = 0

Reorder the terms:
z(-3 + z)(2z + 1) = 0

Reorder the terms:
z(-3 + z)(1 + 2z) = 0

Multiply (-3 + z) * (1 + 2z)
z(-3(1 + 2z) + z(1 + 2z)) = 0
z((1 * -3 + 2z * -3) + z(1 + 2z)) = 0
z((-3 + -6z) + z(1 + 2z)) = 0
z(-3 + -6z + (1 * z + 2z * z)) = 0
z(-3 + -6z + (1z + 2z2)) = 0

Combine like terms: -6z + 1z = -5z
z(-3 + -5z + 2z2) = 0
(-3 * z + -5z * z + 2z2 * z) = 0
(-3z + -5z2 + 2z3) = 0

Solving
-3z + -5z2 + 2z3 = 0

Solving for variable 'z'.

Factor out the Greatest Common Factor (GCF), 'z'.
z(-3 + -5z + 2z2) = 0

Factor a trinomial.
z((-1 + -2z)(3 + -1z)) = 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 '(-1 + -2z)' equal to zero and attempt to solve:

Simplifying
-1 + -2z = 0

Solving
-1 + -2z = 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 + -2z = 0 + 1

Combine like terms: -1 + 1 = 0
0 + -2z = 0 + 1
-2z = 0 + 1

Combine like terms: 0 + 1 = 1
-2z = 1

Divide each side by '-2'.
z = -0.5

Simplifying
z = -0.5
Subproblem 3
Set the factor '(3 + -1z)' equal to zero and attempt to solve:

Simplifying
3 + -1z = 0

Solving
3 + -1z = 0

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

Add '-3' to each side of the equation.
3 + -3 + -1z = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -1z = 0 + -3
-1z = 0 + -3

Combine like terms: 0 + -3 = -3
-1z = -3

Divide each side by '-1'.
z = 3

Simplifying
z = 3
Solution
z = {0, -0.5, 3}