4x^3-12x^2+8x=0

To solve the equation \( 4x^3 - 12x^2 + 8x = 0 \), we can first factor out the common factor. 1. **Factor out the greatest common factor (GCF):** The GCF of the terms is \( 4x \). \[ 4x(x^2 - 3x + 2) = 0 \] 2. **Set each factor equal to zero:** From the factored form \( 4x(x^2 - 3x + 2) = 0 \), we have: \[ 4x = 0 \quad \text{or} \quad x^2 - 3x + 2 = 0 \] - For \( 4x = 0 \): \[ x = 0 \] - For \( x^2 - 3x + 2 = 0 \), we can factor this quadratic equation: \[ (x - 1)(x - 2) = 0 \] Setting each factor to zero gives us: \[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \] \[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \] 3. **Summarize the solutions:** The solutions to the equation \( 4x^3 - 12x^2 + 8x = 0 \) are: \[ x = 0, \quad x = 1, \quad x = 2 \] So the final answer is \( x = 0, 1, 2 \).