What is the answer to z-1/z =5+3i ?

The answer to z-1/z =5+3i is z=-0.14706…+0.08333…i,z=5.14706…+2.91666…i

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z-1/z =5+3i

z - \frac{1}{z} = 5 + 3 i

z = - 0.14706 \dots + 0.08333 \dots i, z = 5.14706 \dots + 2.91666 \dots i

Solution steps

z - \frac{1}{z} = 5 + 3 i

Substitute

z = x + y i

x + y i - \frac{1}{x + y i} = 5 + 3 i

Multiply both sides by

x + y i

x (x + y i) + y i (x + y i) - \frac{1}{x + y i} (x + y i) = 5 (x + y i) + 3 i (x + y i)

Expand

(x^{2} - y^{2} - 1) + 2 i x y = (5 x - 3 y) + i (3 x + 5 y)

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

[x^{2} - y^{2} - 1 = 5 x - 3 y 2 x y = 3 x + 5 y]

[x^{2} - y^{2} - 1 = 5 x - 3 y 2 x y = 3 x + 5 y] : (x = - 0.14706 \dots, x = 5.14706 \dots, y = 0.08333 \dots y = 2.91666 \dots)

Substitute back

z = x + y i

z = - 0.14706 \dots + 0.08333 \dots i, z = 5.14706 \dots + 2.91666 \dots i

\frac{d}{3.5 h} = 100 \frac{mi}{h}

(a - bi) (a + bi + 1) = 11 + 3 i

a (1 - 3 i) + b (4 - i) = - 17 - 4 i

(a + ib)^{2} = i

(1 - i) x + (2 + i) y = 4 + 2 i

What is the answer to z-1/z =5+3i ?
The answer to z-1/z =5+3i is z=-0.14706…+0.08333…i,z=5.14706…+2.91666…i