What is the answer to z^1+z^2=(4+3)+(2+5)i ?

The answer to z^1+z^2=(4+3)+(2+5)i is z=2.44345…+1.18907…i,z=-3.44345…-1.18907…i

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z^1+z^2=(4+3)+(2+5)i

z^{1} + z^{2} = (4 + 3) + (2 + 5) i

z = 2.44345 \dots + 1.18907 \dots i, z = - 3.44345 \dots - 1.18907 \dots i

Solution steps

z^{1} + z^{2} = (4 + 3) + (2 + 5) i

Substitute

z = x + y i

(x + y i)^{1} + (x + y i)^{2} = 4 + 3 + (2 + 5) i

Expand

(x + x^{2} - y^{2}) + i (y + 2 x y) = 7 + 7 i

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

[x + x^{2} - y^{2} = 7 y + 2 x y = 7]

[x + x^{2} - y^{2} = 7 y + 2 x y = 7] : (x = 2.44345 \dots, x = - 3.44345 \dots, y = 1.18907 \dots y = - 1.18907 \dots)

Substitute back

z = x + y i

z = 2.44345 \dots + 1.18907 \dots i, z = - 3.44345 \dots - 1.18907 \dots i

z^{2} - (1 + 4 i) z - (6 - 12 i) = 0

(2 + i) \cdot a + (i - 3) \cdot b = 4

x^{2} - 2 i x + (- 1 + 2 i) = 0

\frac{x + i y}{1 + ( x - i y )} = 3 + 4 i

(i - 5) z + (i - 2) z^{2} + 2 - 2 i = 0

What is the answer to z^1+z^2=(4+3)+(2+5)i ?
The answer to z^1+z^2=(4+3)+(2+5)i is z=2.44345…+1.18907…i,z=-3.44345…-1.18907…i