What is the answer to 3z^2-8iz-3=0 ?

The answer to 3z^2-8iz-3=0 is z=(4-sqrt(7))/3 i,z=(4+sqrt(7))/3 i

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3z^2-8iz-3=0

3 z^{2} - 8 i z - 3 = 0

z = \frac{4 - 7}{3} i, z = \frac{4 + 7}{3} i

Solution steps

3 z^{2} - 8 i z - 3 = 0

Substitute

z = x + y i

3 (x + y i)^{2} - 8 i (x + y i) - 3 = 0

Expand

3 (x + y i)^{2} - 8 i (x + y i) - 3 : (3 x^{2} - 3 y^{2} + 8 y - 3) + i (- 8 x + 6 x y)

(3 x^{2} - 3 y^{2} + 8 y - 3) + i (- 8 x + 6 x y) = 0

Rewrite

0

in standard complex form:

0 + 0 i

(3 x^{2} - 3 y^{2} + 8 y - 3) + i (- 8 x + 6 x y) = 0 + 0 i

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

[3 x^{2} - 3 y^{2} + 8 y - 3 = 0 - 8 x + 6 x y = 0]

[3 x^{2} - 3 y^{2} + 8 y - 3 = 0 - 8 x + 6 x y = 0] : (x = 0, x = 0, y = \frac{4 - 7}{3} y = \frac{4 + 7}{3})

Substitute back

z = x + y i

z = \frac{4 - 7}{3} i, z = \frac{4 + 7}{3} i

(3 - i) (a + bi) = 0

x + i y = y + i x

(2 - 5 i) x + (1 + 3 i) y = 8 - 9 i

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

x^{2} - 12 i x - 1 = 0

What is the answer to 3z^2-8iz-3=0 ?
The answer to 3z^2-8iz-3=0 is z=(4-sqrt(7))/3 i,z=(4+sqrt(7))/3 i