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

given equation: y^2 = x^3 + 2*x - 3*x*y
assuming plotted sketch correct.

y^2 = x^3 + 2*x - 3*x*y

hint:

y^2 + x^2 =1  ==>  y= sqrt( 1 - x^2 )

the x values known.
how can find corresponding y values x values? e.g. known x-values, expect below listed y-values (see plotted sketch):

x= 1 ; y=0.79 x=2 ; y=1.58 x=3 ; y=2.79 x=4 ; y=4.39 x=5 ; y=6.33 x=6 ; y=8.57  x=7 ; y=11.12  x=8 ; y=13.92 x=9 ; y=16.98 x=10 ; y= 20.29

e.g. try find y x=6; y calculated follws:

y^2+x^2=1  ==>  y=sqrt(1 - x^2) = sqrt(1-36) = sqrt(-35) = (0, 5.92i );

thus:

y^2 = x^3 + 2*x - 3*x*y = (228 , -106,49i)  y = sqrt( 228 , -106,49i) = (15.49 , -3.44i)

sadly, calculated y wrong! expect (6, 8.57i). how can find y?

thanks in advance.

just solve 'y'. it's not difficoult, when treat x constant value:

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

it's quadratic equation of:
= -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 may see chart there 2 y matched 1 x. think clear enaugh :)

Breniser