What is the answer to 2z^2+4z+1-isqrt(3)=0 ?

The answer to 2z^2+4z+1-isqrt(3)=0 is z=-0.13397…+0.5i,z=-1.86602…-0.5i

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2z^2+4z+1-isqrt(3)=0

2 z^{2} + 4 z + 1 - i 3 = 0

z = - 0.13397 \dots + 0.5 i, z = - 1.86602 \dots - 0.5 i

Solution steps

2 z^{2} + 4 z + 1 - i 3 = 0

Substitute

z = x + y i

2 (x + y i)^{2} + 4 (x + y i) + 1 - i 3 = 0

Expand

2 (x + y i)^{2} + 4 (x + y i) + 1 - i 3 : (2 x^{2} - 2 y^{2} + 4 x + 1) + i (- 3 + 4 y + 4 x y)

(2 x^{2} - 2 y^{2} + 4 x + 1) + i (- 3 + 4 y + 4 x y) = 0

Rewrite

0

in standard complex form:

0 + 0 i

(2 x^{2} - 2 y^{2} + 4 x + 1) + i (- 3 + 4 y + 4 x y) = 0 + 0 i

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

[2 x^{2} - 2 y^{2} + 4 x + 1 = 0 - 3 + 4 y + 4 x y = 0]

[2 x^{2} - 2 y^{2} + 4 x + 1 = 0 - 3 + 4 y + 4 x y = 0] : (x = - 0.13397 \dots, x = - 1.86602 \dots, y = 0.5 y = - 0.5)

Substitute back

z = x + y i

z = - 0.13397 \dots + 0.5 i, z = - 1.86602 \dots - 0.5 i

- i x^{2} y = x^{2} + y + 4 i

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

z - (\frac{1}{z}) = 3 + 4 i

(5 + 2 i) x + (3 - 5 i) y = 124

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

What is the answer to 2z^2+4z+1-isqrt(3)=0 ?
The answer to 2z^2+4z+1-isqrt(3)=0 is z=-0.13397…+0.5i,z=-1.86602…-0.5i