If the sum of two numbers is $-2$ and the sum of their cubes is $0$, find the two numbers.
Step-by-Step Solution
Key Concept: Use the sum of cubes factorization $x^3 + y^3 = (x+y)^3 - 3xy(x+y)$ to relate the given conditions.
Let the two numbers be $z$ and $y$. Given that their sum is $-2$: $x + y = -2$, so $y = -2 - x$. Also given that the sum of their cubes is $0$: $x^3 + y^3 = 0$. Using the identity $x^3 + y^3 = (x+y)^3 - 3xy(x+y)$, we get $0 = (-2)^3 - 3xy(-2) = -8 + 6xy$, which gives $xy = \frac{4}{3}$. This leads to the quadratic $x^2 + 2x - \frac{4}{3} = 0$. Solving using the given equations simultaneously yields $x = -2, y = 0$ or $x = 0, y = -2$.
Correct Answer: -2, 0