Answer to Question #347049 in Complex Analysis for Images

Question #347049

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


1
Expert's answer
2022-06-07T00:22:31-0400
"a z^3 +b z^2 +c z+d=0\\\\\n2 z^3 - 5 z^2 + z - 5=0"

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

"\\Delta_0=b^2-3ac=19\\\\\n\\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} \n\\!+\\! (350 \\!+ \\!3 \\sqrt{12849})^{1\/3})"

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


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