What is the answer to e^{i pi/3}z^2-2e^{i pi/6}z+2=0 ?

The answer to e^{i pi/3}z^2-2e^{i pi/6}z+2=0 is

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e^{i pi/3}z^2-2e^{i pi/6}z+2=0

e^{i \frac{π}{3}} z^{2} - 2 e^{i \frac{π}{6}} z + 2 = 0

z = 0.36602 \dots - \frac{- 2 + 3 \cdot 0.73205 \dots + 22.14359 \dots - 3 \cdot 5.85640 \dots}{2} i, z = 1.36602 \dots + \frac{2 - 3 \cdot 2.73205 \dots + 49.85640 \dots - 3 \cdot 21.85640 \dots}{2} i

Solution steps

e^{i \frac{π}{3}} z^{2} - 2 e^{i \frac{π}{6}} z + 2 = 0

Substitute

z = x + y i

e^{i \frac{π}{3}} (x + y i)^{2} - 2 e^{i \frac{π}{6}} (x + y i) + 2 = 0

Expand

e^{i \frac{π}{3}} (x + y i)^{2} - 2 e^{i \frac{π}{6}} (x + y i) + 2 : \frac{4 + x ^{2} - y ^{2} + 2 y - 2 3 x - 2 3 x y}{2} + i \frac{- 2 x + 3 x ^{2} - 3 y ^{2} + 2 x y - 2 3 y}{2}

\frac{4 + x ^{2} - y ^{2} + 2 y - 2 3 x - 2 3 x y}{2} + i \frac{- 2 x + 3 x ^{2} - 3 y ^{2} + 2 x y - 2 3 y}{2} = 0

Rewrite

0

in standard complex form:

0 + 0 i

\frac{4 + x ^{2} - y ^{2} + 2 y - 2 3 x - 2 3 x y}{2} + i \frac{- 2 x + 3 x ^{2} - 3 y ^{2} + 2 x y - 2 3 y}{2} = 0 + 0 i

Complex numbers can be equal only if their real and imaginary parts are equalRewrite as system of equations:

[\frac{4 + x ^{2} - y ^{2} + 2 y - 2 3 x - 2 3 x y}{2} = 0 \frac{- 2 x + 3 x ^{2} - 3 y ^{2} + 2 x y - 2 3 y}{2} = 0]

[\frac{4 + x ^{2} - y ^{2} + 2 y - 2 3 x - 2 3 x y}{2} = 0 \frac{- 2 x + 3 x ^{2} - 3 y ^{2} + 2 x y - 2 3 y}{2} = 0] : x = 0.36602 \dots, x = 1.36602 \dots, y = - \frac{- 2 + 3 \cdot 0.73205 \dots + 22.14359 \dots - 3 \cdot 5.85640 \dots}{2} y = \frac{2 - 3 \cdot 2.73205 \dots + 49.85640 \dots - 3 \cdot 21.85640 \dots}{2}

Substitute back

z = x + y i

z = 0.36602 \dots - \frac{- 2 + 3 \cdot 0.73205 \dots + 22.14359 \dots - 3 \cdot 5.85640 \dots}{2} i, z = 1.36602 \dots + \frac{2 - 3 \cdot 2.73205 \dots + 49.85640 \dots - 3 \cdot 21.85640 \dots}{2} i

\frac{1}{x ^{- 1} + y ^{- 1}}

lo g_{5} (\frac{1}{5})

5 z + 6 \leq - 7 z + 7

- 8 x^{2} - 20 x - 12

∣ 4 x - 5 ∣ \geq 4

What is the answer to e^{i pi/3}z^2-2e^{i pi/6}z+2=0 ?
The answer to e^{i pi/3}z^2-2e^{i pi/6}z+2=0 is