Assume z = x + iy, then find a complex number z satisfying the given equation. d. 2z8 – 2z4 + 1 = 0

Answer: complex equations has n number of solutions, been n the equation degree. In this case:[tex]Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i11,25°}[/tex][tex]Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i101,25°}[/tex][tex]Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i191,25°}[/tex][tex]Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i281,25°}[/tex][tex]Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i78,75°}[/tex][tex]Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i168,75°}[/tex][tex]Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i258,75°}[/tex][tex]Z=\frac{\sqrt[8]{2} }{\sqrt[4]{2}} e^{i348,75°}[/tex]Step-by-step explanation:I start with a variable substitution:[tex]Z^{4} = X[/tex]Then:[tex]2X^{2}-2X+1=0[/tex]Solving the quadratic equation:[tex]X_{1} =\frac{2+\sqrt{4-4*2*1} }{2*2} \\X_{2} =\frac{2-\sqrt{4-4*2*1} }{2*2}[/tex][tex]X=\left \{ {{0,5+0,5i} \atop {0,5-0,5i}} \right.[/tex]Replacing for the original variable: [tex]Z=\sqrt[4]{0,5+0,5i}[/tex]or [tex]Z=\sqrt[4]{0,5-0,5i}[/tex]Remembering that complex numbers can be written as:[tex]Z=a+ib=|Z|e^{ic}[/tex]Using this:[tex]Z=\left \{ {{{\frac{\sqrt{2}}{2} e^{i45°} } \atop {{\frac{\sqrt{2}}{2} e^{i-45°} }} \right.[/tex]Solving for the modulus and the angle: [tex]Z=\left \{ {{\sqrt[4]{\frac{\sqrt{2}}{2} e^{i45}} = \sqrt[4]{\frac{\sqrt{2}}{2} } \sqrt[4]{e^{i45}} } \atop {\sqrt[4]{\frac{\sqrt{2}}{2} e^{i-45}} = \sqrt[4]{\frac{\sqrt{2}}{2} } \sqrt[4]{e^{i-45}} }} \right.[/tex]The possible angle respond to:[tex]RAng_{12...n} =\frac{Ang +360*(i-1)}{n}[/tex]Been "RAng" the resultant angle, "Ang" the original angle, "n" the degree of the root and "i" a value between 1 and "n"In this case n=4 with 2 different angles: Ang = 45º and Ang = 315ºObtaining 8 different angles, therefore 8 different solutions.