What is the answer to (a+i)^2=8-6i ?

The answer to (a+i)^2=8-6i is a=3-2i,a=-3

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(a+i)^2=8-6i

(a + i)^{2} = 8 - 6 i

a = 3 - 2 i, a = - 3

Solution steps

(a + i)^{2} = 8 - 6 i

Substitute

a = x + y i

(x + y i + i)^{2} = 8 - 6 i

Expand

(x + y i + i)^{2} : (x^{2} - 1 - y^{2} - 2 y) + i (2 x + 2 x y)

(x^{2} - 1 - y^{2} - 2 y) + i (2 x + 2 x y) = 8 - 6 i

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

[x^{2} - 1 - y^{2} - 2 y = 8 2 x + 2 x y = - 6]

[x^{2} - 1 - y^{2} - 2 y = 8 2 x + 2 x y = - 6] : (x = 3, x = - 3, y = - 2 y = 0)

Substitute back

a = x + y i

a = 3 - 2 i, a = - 3

(x + i y) (3 - i) = - i

(x + 4 i) (x - 4 i) = 0

z = \frac{c - 2 i}{2 - i}

\frac{z + 2}{z - 2} = 3 + i^{27}

∣ z ∣^{2} + 1 + 12 i = 6 z