How to find Y for corresponding X values (Implicit function, Complex number)

Question

Given is the equation: Y^2 = X^3 + 2*X - 3*X*Y
Assuming the plotted sketch is correct.

Y^2 = X^3 + 2*X - 3*X*Y

Hint:

Answer 1

just solve it by 'y'. It's not that difficoult, when you treat x like constant value:

y^2 = x^3 + 2x - 3xy
0 = (-1)y^2 + (-3x)y + (x^3 + 2x)

it's Quadratic equation of:
a = -1
b = -3x
c = x^3 + 2x

y1 = (-(-3x) - sqr((-3x)^2 - 4(-1)(x^3+2x)))/2*(-1)
y2 = (-(-3x) + sqr((-3x)^2 - 4(-1)(x^3+2x)))/2*(-1)

finally:

d = x(9*x+4*x^2+8)
y1 = (3x+sqr(d))/(-2)
y2 = (3x-sqr(d))/(-2)

eg.

for x = 6
y1 = -26,5784
y2 = 8,578396

as you may see from the chart there are always two y matched to one x. I think that is clear enaugh :)

Answer 2

Have you used a math library with support for complex numbers ? MathJs is one. See this SO answer.