Find Two Numbers 824Bad

1. **State the problem:** We need to find two numbers whose product is $-42$ and whose sum is $-11$. 2. **Use the formulas:** Let the two numbers be $x$ and $y$. Then: $$x \times y = -42$$ $$x + y = -11$$ 3. **Express one variable:** From the sum equation, express $y$ as: $$y = -11 - x$$ 4. **Substitute into the product equation:** $$x \times (-11 - x) = -42$$ which simplifies to: $$-11x - x^2 = -42$$ 5. **Rewrite as a quadratic equation:** $$-x^2 - 11x + 42 = 0$$ Multiply both sides by $-1$ to simplify: $$\cancel{-}1 \times (-x^2 - 11x + 42) = \cancel{-}1 \times 0$$ $$x^2 + 11x - 42 = 0$$ 6. **Factor the quadratic:** We look for two numbers that multiply to $-42$ and add to $11$. The factors of $-42$ are $(1, -42), (2, -21), (3, -14), (6, -7)$ and their negatives. The pair that adds to $11$ is $14$ and $-3$ because $14 + (-3) = 11$ and $14 \times (-3) = -42$. 7. **Write the factors:** $$(x + 14)(x - 3) = 0$$ 8. **Solve for $x$:** $$x + 14 = 0 \Rightarrow x = -14$$ $$x - 3 = 0 \Rightarrow x = 3$$ 9. **Find corresponding $y$ values:** If $x = -14$, then $y = -11 - (-14) = 3$. If $x = 3$, then $y = -11 - 3 = -14$. 10. **Answer:** The two numbers are $-14$ and $3$. **Final answer:** $-14$ and $3$