What is the answer to x^2-4x+13=0.2+3i ?

The answer to x^2-4x+13=0.2+3i is x=1.50134…-3.00809…i,x=2.49865…+3.00809…i

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x^2-4x+13=0.2+3i

x^{2} - 4 x + 13 = 0.2 + 3 i

x = 1.50134 \dots - 3.00809 \dots i, x = 2.49865 \dots + 3.00809 \dots i

Solution steps

x^{2} - 4 x + 13 = 0.2 + 3 i

Substitute

x = a + bi

(a + bi)^{2} - 4 (a + bi) + 13 = 0.2 + 3 i

Expand

(a + bi)^{2} - 4 (a + bi) + 13 : (a^{2} - b^{2} - 4 a + 13) + i (- 4 b + 2 ab)

(a^{2} - b^{2} - 4 a + 13) + i (- 4 b + 2 ab) = 0.2 + 3 i

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

[a^{2} - b^{2} - 4 a + 13 = 0.2 - 4 b + 2 ab = 3]

[a^{2} - b^{2} - 4 a + 13 = 0.2 - 4 b + 2 ab = 3] : (a = 1.50134 \dots, a = 2.49865 \dots, b = - 3.00809 \dots b = 3.00809 \dots)

Substitute back

x = a + bi

x = 1.50134 \dots - 3.00809 \dots i, x = 2.49865 \dots + 3.00809 \dots i

- 3 x^{2} - 298 = - 60 x - 1

w (2 w - 11) - 8 = 0

5 r^{2} + 15 r + 7 = 0

20 x^{2} - 20 x = 40

- x^{2} = - 15 - 2 x

What is the answer to x^2-4x+13=0.2+3i ?
The answer to x^2-4x+13=0.2+3i is x=1.50134…-3.00809…i,x=2.49865…+3.00809…i