What is the answer to z^2-(5+4i)z+(3-5i)=0 ?

The answer to z^2-(5+4i)z+(3-5i)=0 is z=-0.17102…-0.80791…i,z=5.17102…+4.80791…i

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z^2-(5+4i)z+(3-5i)=0

z^{2} - (5 + 4 i) z + (3 - 5 i) = 0

z = - 0.17102 \dots - 0.80791 \dots i, z = 5.17102 \dots + 4.80791 \dots i

Solution steps

z^{2} - (5 + 4 i) z + (3 - 5 i) = 0

Substitute

z = x + y i

(x + y i)^{2} - (5 + 4 i) (x + y i) + 3 - 5 i = 0

Expand

(x + y i)^{2} - (5 + 4 i) (x + y i) + 3 - 5 i : (x^{2} - y^{2} - 5 x + 4 y + 3) + i (- 5 - 4 x - 5 y + 2 x y)

(x^{2} - y^{2} - 5 x + 4 y + 3) + i (- 5 - 4 x - 5 y + 2 x y) = 0

Rewrite

0

in standard complex form:

0 + 0 i

(x^{2} - y^{2} - 5 x + 4 y + 3) + i (- 5 - 4 x - 5 y + 2 x y) = 0 + 0 i

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

[x^{2} - y^{2} - 5 x + 4 y + 3 = 0 - 5 - 4 x - 5 y + 2 x y = 0]

[x^{2} - y^{2} - 5 x + 4 y + 3 = 0 - 5 - 4 x - 5 y + 2 x y = 0] : (x = - 0.17102 \dots, x = 5.17102 \dots, y = - 0.80791 \dots y = 4.80791 \dots)

Substitute back

z = x + y i

z = - 0.17102 \dots - 0.80791 \dots i, z = 5.17102 \dots + 4.80791 \dots i

x^{2} + 3 x + 2 - i = 0

x^{2} - 3 + 4 i = 0

- 8.4 x i - 3.15 = - x^{2}

i x = x^{2} - 2 x + 2

x^{2} + 2 x i + 1 = 0

What is the answer to z^2-(5+4i)z+(3-5i)=0 ?
The answer to z^2-(5+4i)z+(3-5i)=0 is z=-0.17102…-0.80791…i,z=5.17102…+4.80791…i