Questions: Solve for all solutions by factoring: 5x^3 = 20x^2 + 60x (Put in standard form, factor, then solve.)

Solution Steps

To solve the equation \(5x^3 = 20x^2 + 60x\) by factoring, we first need to rewrite the equation in standard form by moving all terms to one side of the equation. Then, we factor out the greatest common factor and solve for the values of \(x\) that satisfy the equation.

We start with the equation: \[ 5x^3 = 20x^2 + 60x \] Rearranging it to standard form gives: \[ 5x^3 - 20x^2 - 60x = 0 \]

Next, we factor out the greatest common factor from the left-hand side: \[ 5x(x^2 - 4x - 12) = 0 \] We can further factor the quadratic \(x^2 - 4x - 12\): \[ 5x(x - 6)(x + 2) = 0 \]

Setting each factor equal to zero gives us the solutions:

1. \(5x = 0 \Rightarrow x = 0\)
2. \(x - 6 = 0 \Rightarrow x = 6\)
3. \(x + 2 = 0 \Rightarrow x = -2\)

Final Answer