Final answer:
To solve the equation 12x³ = 2x² + 24x, factor out a 2x and then set each factor equal to zero. The solutions are x = 0, x = 4, and x = -3.
Step-by-step explanation:
To solve the equation 12x³ = 2x² + 24x, we can start by setting the equation equal to zero: 12x³ - 2x² - 24x = 0. Then, we can factor out a 2x from each term: 2x(x² - x - 12) = 0. Next, we can factor the quadratic expression x² - x - 12 by finding two numbers whose product is -12 and whose sum is -1 (the coefficient of the x term). The numbers are -4 and 3.
Therefore, the factored form of the equation is 2x(x - 4)(x + 3) = 0. From here, we can use the zero product property, which states that if the product of two or more factors is zero, then at least one of the factors must be zero.
So, we set each factor equal to zero and solve for x. The solutions are x = 0, x = 4, and x = -3.