1. **State the problem:** Solve the cubic equation $$x^3 - 2x - 5 = 0$$ for $x$. 2. **Recall the formula:** For cubic equations of the form $$x^3 + ax^2 + bx + c = 0$$, we can use methods like factoring, the Rational Root Theorem, or Cardano's formula. Here, $a=0$, $b=-2$, and $c=-5$. 3. **Check for rational roots using the Rational Root Theorem:** Possible rational roots are factors of $c$ over factors of leading coefficient, i.e., $\pm1, \pm5$. 4. **Test $x=1$:** $$1^3 - 2(1) - 5 = 1 - 2 - 5 = -6 \neq 0$$ 5. **Test $x=-1$:** $$(-1)^3 - 2(-1) - 5 = -1 + 2 - 5 = -4 \neq 0$$ 6. **Test $x=5$:** $$5^3 - 2(5) - 5 = 125 - 10 - 5 = 110 \neq 0$$ 7. **Test $x=-5$:** $$(-5)^3 - 2(-5) - 5 = -125 + 10 - 5 = -120 \neq 0$$ No rational roots found. 8. **Use Cardano's formula for depressed cubic:** The equation is already depressed since $a=0$. 9. **Set:** $$x = u + v$$ Then $$x^3 - 2x - 5 = (u+v)^3 - 2(u+v) - 5 = 0$$ 10. **Expand:** $$u^3 + v^3 + 3uv(u+v) - 2(u+v) - 5 = 0$$ 11. **Group terms:** $$u^3 + v^3 + (3uv - 2)(u+v) - 5 = 0$$ 12. **Set:** $$3uv - 2 = 0 \Rightarrow uv = \frac{2}{3}$$ 13. **Then:** $$u^3 + v^3 - 5 = 0 \Rightarrow u^3 + v^3 = 5$$ 14. **Let:** $$s = u^3, t = v^3$$ Then $$s + t = 5$$ and $$st = (uv)^3 = \left(\frac{2}{3}\right)^3 = \frac{8}{27}$$ 15. **Solve quadratic for $s$:** $$z^2 - 5z + \frac{8}{27} = 0$$ 16. **Calculate discriminant:** $$\Delta = 5^2 - 4 \times 1 \times \frac{8}{27} = 25 - \frac{32}{27} = \frac{675 - 32}{27} = \frac{643}{27}$$ 17. **Roots:** $$z = \frac{5 \pm \sqrt{\frac{643}{27}}}{2} = \frac{5 \pm \frac{\sqrt{643}}{\sqrt{27}}}{2} = \frac{5 \pm \frac{\sqrt{643}}{3\sqrt{3}}}{2}$$ 18. **Approximate:** $$\sqrt{643} \approx 25.35$$ so $$z \approx \frac{5 \pm \frac{25.35}{5.196}}{2} = \frac{5 \pm 4.88}{2}$$ 19. **Calculate roots:** - $$z_1 \approx \frac{5 + 4.88}{2} = \frac{9.88}{2} = 4.94$$ - $$z_2 \approx \frac{5 - 4.88}{2} = \frac{0.12}{2} = 0.06$$ 20. **Then:** $$u = \sqrt[3]{4.94} \approx 1.7$$ and $$v = \sqrt[3]{0.06} \approx 0.39$$ 21. **Solution:** $$x = u + v \approx 1.7 + 0.39 = 2.09$$ 22. **Check:** $$2.09^3 - 2(2.09) - 5 \approx 9.11 - 4.18 - 5 = -0.07 \approx 0$$ (close to zero, confirming solution). **Final answer:** $$x \approx 2.09$$