Solve the Cubic Equation: x³ - 5x² = -6x Step-by-Step

To solve the equation $x^3 - 5x^2 = -6x$, we'll use factoring:

• Step 1: Start by rearranging the equation: $x^3 - 5x^2 + 6x = 0$.
• Step 2: Factor out the greatest common factor, which is $x$:

$x(x^2 - 5x + 6) = 0$

• Step 3: The quadratic $x^2 - 5x + 6$ can be factored further:

$x^2 - 5x + 6 = (x - 2)(x - 3)$

• Step 4: Substitute back into the equation:

$x(x - 2)(x - 3) = 0$

• Step 5: Use the zero-product property to find the solutions:

Set each factor to zero: $x = 0$, $x - 2 = 0$, and $x - 3 = 0$.

• Step 6: Solve each equation:

$x = 0$

$x = 2$

$x = 3$

Therefore, the solutions to the equation $x^3 - 5x^2 = -6x$ are $x = 0, 2, 3$.