What is the answer to (5+2i)/z =2-i ?

The answer to (5+2i)/z =2-i is z= 8/5+9/5 i

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(5+2i)/z =2-i

\frac{5 + 2 i}{z} = 2 - i

z = \frac{8}{5} + \frac{9}{5} i

Solution steps

\frac{5 + 2 i}{z} = 2 - i

Substitute

z = x + y i

\frac{5 + 2 i}{x + y i} = 2 - i

Multiply both sides by

x + y i

\frac{5 + 2 i}{x + y i} (x + y i) = 2 (x + y i) - i (x + y i)

Expand

5 + 2 i = (2 x + y) + i (- x + 2 y)

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

[5 = 2 x + y 2 = - x + 2 y]

[5 = 2 x + y 2 = - x + 2 y] : x = \frac{8}{5}, y = \frac{9}{5}

Substitute back

z = x + y i

z = \frac{8}{5} + \frac{9}{5} i

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

(x - y) + 5 i = 2 + y i

(z + i) (z - 2 i) = z^{2} + i z + 2

(u - i) (4 + i v) = 11 + 2 i

z^{2} - 7 i z - 12 = 0

What is the answer to (5+2i)/z =2-i ?
The answer to (5+2i)/z =2-i is z= 8/5+9/5 i