1. **State the problem:** Solve the cubic equation $$-x^3 + 6x^2 + 3x - 13 = 0$$ for $x$. 2. **Rewrite the equation:** Multiply both sides by $-1$ to simplify the leading coefficient: $$x^3 - 6x^2 - 3x + 13 = 0$$ 3. **Use Rational Root Theorem:** Possible rational roots are factors of 13 over factors of 1, i.e., $\pm1, \pm13$. 4. **Test possible roots:** - For $x=1$: $1 - 6 - 3 + 13 = 5 \neq 0$ - For $x=-1$: $-1 - 6 + 3 + 13 = 9 \neq 0$ - For $x=13$: $2197 - 1014 - 39 + 13 = 1157 \neq 0$ - For $x=-13$: $-2197 - 1014 + 39 + 13 = -3159 \neq 0$ No rational roots found. 5. **Use numerical methods or Cardano's formula:** The cubic can be solved using the depressed cubic method or numerical approximation. 6. **Depress the cubic:** Let $x = y + \frac{6}{3} = y + 2$ to eliminate the $x^2$ term. Substitute: $$ (y+2)^3 - 6(y+2)^2 - 3(y+2) + 13 = 0 $$ Expand: $$ y^3 + 6y^2 + 12y + 8 - 6(y^2 + 4y + 4) - 3y - 6 + 13 = 0 $$ Simplify: $$ y^3 + 6y^2 + 12y + 8 - 6y^2 - 24y - 24 - 3y - 6 + 13 = 0 $$ $$ y^3 + (6y^2 - 6y^2) + (12y - 24y - 3y) + (8 - 24 - 6 + 13) = 0 $$ $$ y^3 - 15y - 9 = 0 $$ 7. **Solve depressed cubic:** $y^3 + py + q = 0$ with $p = -15$, $q = -9$. 8. **Calculate discriminant:** $$ \Delta = \left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3 = \left(-\frac{9}{2}\right)^2 + \left(-5\right)^3 = \frac{81}{4} - 125 = -\frac{419}{4} < 0 $$ Since $\Delta < 0$, there are three real roots. 9. **Use trigonometric solution:** $$ y = 2\sqrt{-\frac{p}{3}} \cos\left(\frac{1}{3} \arccos\left(\frac{3q}{2p} \sqrt{-\frac{3}{p}}\right) - \frac{2\pi k}{3}\right), \quad k=0,1,2 $$ Calculate: $$ \sqrt{-\frac{p}{3}} = \sqrt{5} $$ $$ \theta = \arccos\left(\frac{3(-9)}{2(-15)} \sqrt{\frac{3}{15}}\right) = \arccos\left(\frac{-27}{-30} \sqrt{0.2}\right) = \arccos(0.9 \times 0.4472) = \arccos(0.4025) \approx 1.155 $$ 10. **Roots for $y$:** $$ y_k = 2\sqrt{5} \cos\left(\frac{1.155}{3} - \frac{2\pi k}{3}\right), k=0,1,2 $$ Calculate each root: - $y_0 = 2\sqrt{5} \cos(0.385) \approx 4.472 \times 0.927 = 4.146$ - $y_1 = 2\sqrt{5} \cos(0.385 - 2.094) = 2\sqrt{5} \cos(-1.709) \approx 4.472 \times (-0.139) = -0.621$ - $y_2 = 2\sqrt{5} \cos(0.385 - 4.188) = 2\sqrt{5} \cos(-3.803) \approx 4.472 \times (-0.788) = -3.525$ 11. **Convert back to $x$:** $$ x = y + 2 $$ So, - $x_0 = 4.146 + 2 = 6.146$ - $x_1 = -0.621 + 2 = 1.379$ - $x_2 = -3.525 + 2 = -1.525$ **Final answer:** The three real solutions are approximately $$ x \approx 6.146, 1.379, -1.525 $$