What is the answer to x^2-4x+40=2-6i ?

The answer to x^2-4x+40=2-6i is x=1.48748…+5.85343…i,x=2.51251…-5.85343…i

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x^2-4x+40=2-6i

x^{2} - 4 x + 40 = 2 - 6 i

x = 1.48748 \dots + 5.85343 \dots i, x = 2.51251 \dots - 5.85343 \dots i

Solution steps

x^{2} - 4 x + 40 = 2 - 6 i

Substitute

x = a + bi

(a + bi)^{2} - 4 (a + bi) + 40 = 2 - 6 i

Expand

(a + bi)^{2} - 4 (a + bi) + 40 : (a^{2} - b^{2} - 4 a + 40) + i (- 4 b + 2 ab)

(a^{2} - b^{2} - 4 a + 40) + i (- 4 b + 2 ab) = 2 - 6 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 + 40 = 2 - 4 b + 2 ab = - 6]

[a^{2} - b^{2} - 4 a + 40 = 2 - 4 b + 2 ab = - 6] : (a = 1.48748 \dots, a = 2.51251 \dots, b = 5.85343 \dots b = - 5.85343 \dots)

Substitute back

x = a + bi

x = 1.48748 \dots + 5.85343 \dots i, x = 2.51251 \dots - 5.85343 \dots i

(x + i y) (2 + 1) = 3 - i

3 x + 8 i = 12 + 12

(x + y) + (x - y) \cdot i = 5 + 5 \cdot i

5 x + 6 i = - 8 + 6

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

What is the answer to x^2-4x+40=2-6i ?
The answer to x^2-4x+40=2-6i is x=1.48748…+5.85343…i,x=2.51251…-5.85343…i