(2x-1)x(x+1)=0

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

Simplifying
(2x + -1) * x(x + 1) = 0

Reorder the terms:
(-1 + 2x) * x(x + 1) = 0

Reorder the terms:
(-1 + 2x) * x(1 + x) = 0

Reorder the terms for easier multiplication:
x(-1 + 2x)(1 + x) = 0

Multiply (-1 + 2x) * (1 + x)
x(-1(1 + x) + 2x * (1 + x)) = 0
x((1 * -1 + x * -1) + 2x * (1 + x)) = 0
x((-1 + -1x) + 2x * (1 + x)) = 0
x(-1 + -1x + (1 * 2x + x * 2x)) = 0
x(-1 + -1x + (2x + 2x2)) = 0

Combine like terms: -1x + 2x = 1x
x(-1 + 1x + 2x2) = 0
(-1 * x + 1x * x + 2x2 * x) = 0
(-1x + 1x2 + 2x3) = 0

Solving
-1x + 1x2 + 2x3 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), 'x'.
x(-1 + x + 2x2) = 0

Factor a trinomial.
x((-1 + -1x)(1 + -2x)) = 0

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

Simplifying
x = 0

Solving
x = 0

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

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

Simplifying
-1 + -1x = 0

Solving
-1 + -1x = 0

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

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

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

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

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

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

Simplifying
1 + -2x = 0

Solving
1 + -2x = 0

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

Add '-1' to each side of the equation.
1 + -1 + -2x = 0 + -1

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

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

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

Simplifying
x = 0.5
Solution
x = {0, -1, 0.5}