Question

solve the cubic equation 2z³ -5z²+z-5=0

Answer 1

$$a z^3 +b z^2 +c z+d=0\\ 2 z^3 - 5 z^2 + z - 5=0$$

$$a=2,\; b=-5,\; c=1,\; d=-5$$

$$\Delta_0=b^2-3ac=19\\ \Delta_1=2b^3-9abc+27a^2d=-700$$

$$C=\sqrt[3]{\frac{\Delta_1+\sqrt{\Delta_1^2-4\Delta_0^3}}{2}}=\sqrt[3]{-350+3\sqrt{12849}}$$

$$z_k=-\frac{1}{3a}(b+\xi^kC+\frac{\Delta_0}{\xi^kC}),\; k=0,1,2$$

$$\xi=\frac{-1+i\sqrt{3}}{2}$$

$$z_0\!=\!\frac{5}{6}\!+\!\frac{1}{6} ( (350\! -\! 3 \sqrt{12849})^{1/3} \!+\! (350 \!+ \!3 \sqrt{12849})^{1/3})$$

$$z_1=\frac{5}{6} -\frac{1}{12} (1 + i \sqrt{3}) (350 - 3 \sqrt{12849})^{1/3} \\ - \frac{1}{12} (1 - i \sqrt{3}) (350 + 3 \sqrt{12849})^{1/3}$$

$$z_2=\frac{5}{6} -\frac{1}{12} (1 - i \sqrt{3}) (350 - 3 \sqrt{12849})^{1/3} \\ - \frac{1}{12} (1+ i \sqrt{3}) (350 + 3 \sqrt{12849})^{1/3}$$