What is the answer to i(x)=3-4/(2-x) ?

The answer to i(x)=3-4/(2-x) is x=0.58120…-0.30610…i,x=1.41879…-2.69389…i

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i(x)=3-4/(2-x)

i (x) = 3 - \frac{4}{2 - x}

x = 0.58120 \dots - 0.30610 \dots i, x = 1.41879 \dots - 2.69389 \dots i

Solution steps

i (x) = 3 - \frac{4}{2 - x}

Substitute

x = a + bi

i (a + bi) = 3 - \frac{4}{2 - ( a + bi )}

Multiply both sides by

2 - a - ib

i (a + bi) (2 - a - ib) = 3 (2 - a - ib) - \frac{4}{2 - ( a + bi )} (2 - a - ib)

Expand

(2 ab - 2 b) + i (- a^{2} + b^{2} + 2 a) = (2 - 3 a) - 3 ib

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

[2 ab - 2 b = 2 - 3 a - a^{2} + b^{2} + 2 a = - 3 b]

[2 ab - 2 b = 2 - 3 a - a^{2} + b^{2} + 2 a = - 3 b] : (a = 0.58120 \dots, a = 1.41879 \dots, b = - 0.30610 \dots b = - 2.69389 \dots)

Substitute back

x = a + bi

x = 0.58120 \dots - 0.30610 \dots i, x = 1.41879 \dots - 2.69389 \dots i

x + y i = 9 + 2 i

z^{2} + (- 2 + 3 i) z - 2 - 4 i = 0

z^{2} = 1 + 3 i

(x + i y) (- i) = 14 i

1 = ai + b, 2 - i = - a + b

What is the answer to i(x)=3-4/(2-x) ?
The answer to i(x)=3-4/(2-x) is x=0.58120…-0.30610…i,x=1.41879…-2.69389…i