1. **State the problem:** Solve the equation $y^2 + y^3 = 14$ for $y$. 2. **Rewrite the equation:** The equation can be written as $$y^3 + y^2 = 14$$ 3. **Isolate terms:** We want to find values of $y$ that satisfy this cubic equation. 4. **Try to find rational roots:** Use the Rational Root Theorem to test possible roots among factors of 14: $\pm1, \pm2, \pm7, \pm14$. 5. **Test $y=2$:** $$2^3 + 2^2 = 8 + 4 = 12 \neq 14$$ 6. **Test $y=1$:** $$1^3 + 1^2 = 1 + 1 = 2 \neq 14$$ 7. **Test $y= -1$:** $$(-1)^3 + (-1)^2 = -1 + 1 = 0 \neq 14$$ 8. **Test $y= 7$:** $$7^3 + 7^2 = 343 + 49 = 392 \neq 14$$ 9. **Test $y= -2$:** $$(-2)^3 + (-2)^2 = -8 + 4 = -4 \neq 14$$ 10. Since no simple rational root works, use substitution or numerical methods. 11. **Rewrite as:** $$y^3 + y^2 - 14 = 0$$ 12. **Use the cubic formula or approximate numerically:** 13. **Approximate root:** By testing values between 2 and 3: - At $y=2.5$: $$2.5^3 + 2.5^2 = 15.625 + 6.25 = 21.875 > 14$$ - At $y=2$: $$12 < 14$$ 14. So root lies between 2 and 2.5. 15. **Try $y=2.2$:** $$2.2^3 + 2.2^2 = 10.648 + 4.84 = 15.488 > 14$$ 16. **Try $y=2.1$:** $$2.1^3 + 2.1^2 = 9.261 + 4.41 = 13.671 < 14$$ 17. Root is between 2.1 and 2.2, closer to 2.15. 18. **Final approximate solution:** $$y \approx 2.14$$ **Answer:** The solution to $y^2 + y^3 = 14$ is approximately $y \approx 2.14$.