What is the answer to 5+(x^2-6x+9)i=0 ?

The answer to 5+(x^2-6x+9)i=0 is x=(-sqrt(10)+6)/2-sqrt(5/2)i,x=(sqrt(10)+6)/2+sqrt(5/2)i

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5+(x^2-6x+9)i=0

5 + (x^{2} - 6 x + 9) i = 0

x = \frac{- 10 + 6}{2} - \frac{5}{2} i, x = \frac{10 + 6}{2} + \frac{5}{2} i

Solution steps

5 + (x^{2} - 6 x + 9) i = 0

Substitute

x = a + bi

5 + ((a + bi)^{2} - 6 (a + bi) + 9) i = 0

Expand

5 + ((a + bi)^{2} - 6 (a + bi) + 9) i : (5 - 2 ab + 6 b) + i (9 + a^{2} - b^{2} - 6 a)

(5 - 2 ab + 6 b) + i (9 + a^{2} - b^{2} - 6 a) = 0

Rewrite

0

in standard complex form:

0 + 0 i

(5 - 2 ab + 6 b) + i (9 + a^{2} - b^{2} - 6 a) = 0 + 0 i

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

[5 - 2 ab + 6 b = 0 9 + a^{2} - b^{2} - 6 a = 0]

[5 - 2 ab + 6 b = 0 9 + a^{2} - b^{2} - 6 a = 0] : a = \frac{- 10 + 6}{2}, a = \frac{10 + 6}{2}, b = - \frac{5}{2} b = \frac{5}{2}

Substitute back

x = a + bi

x = \frac{- 10 + 6}{2} - \frac{5}{2} i, x = \frac{10 + 6}{2} + \frac{5}{2} i

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

(a - bi) ((a + bi) + 1) = 11 + 3 i

x - i y = 2 + 5 i

(4 + 8 i) (x + y i) = 52 + 4 i

∣ z - 4 + 3 i ∣ = 5

What is the answer to 5+(x^2-6x+9)i=0 ?
The answer to 5+(x^2-6x+9)i=0 is x=(-sqrt(10)+6)/2-sqrt(5/2)i,x=(sqrt(10)+6)/2+sqrt(5/2)i