Solve the equation x ^ 3 - 12 = - 3x ^ 2 + 4x . What is the smallest x-value in the solution set ?

Answer 1

To solve the equation x^3 - 12 = -3x^2 + 4x, we need to rearrange it to set it equal to zero:

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

Now, let's factor out the common factors:

x(x^2 - 3x + 4) - 12 = 0

Now, let's solve for x by factoring the quadratic equation inside the parentheses:

x(x - 2)(x - 2) - 12 = 0

Setting each factor equal to zero:

x = 0, x = 2

Therefore, the solution set is x = {0, 2}. The smallest x-value in the solution set is 0.