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2x^2+3y^2 equation

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Numerical solution:

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The solution

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   2      2    
2*x  + 3*y  = 0
$$2 x^{2} + 3 y^{2} = 0$$
Detail solution
This equation is of the form
a*y^2 + b*y + c = 0

A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$y_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$y_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 3$$
$$b = 0$$
$$c = 2 x^{2}$$
, then
D = b^2 - 4 * a * c = 

(0)^2 - 4 * (3) * (2*x^2) = -24*x^2

The equation has two roots.
y1 = (-b + sqrt(D)) / (2*a)

y2 = (-b - sqrt(D)) / (2*a)

or
$$y_{1} = \frac{\sqrt{6} \sqrt{- x^{2}}}{3}$$
$$y_{2} = - \frac{\sqrt{6} \sqrt{- x^{2}}}{3}$$
Vieta's Theorem
rewrite the equation
$$2 x^{2} + 3 y^{2} = 0$$
of
$$a y^{2} + b y + c = 0$$
as reduced quadratic equation
$$y^{2} + \frac{b y}{a} + \frac{c}{a} = 0$$
$$\frac{2 x^{2}}{3} + y^{2} = 0$$
$$p y + q + y^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = \frac{2 x^{2}}{3}$$
Vieta Formulas
$$y_{1} + y_{2} = - p$$
$$y_{1} y_{2} = q$$
$$y_{1} + y_{2} = 0$$
$$y_{1} y_{2} = \frac{2 x^{2}}{3}$$
The graph
Rapid solution [src]
       ___             ___      
     \/ 6 *im(x)   I*\/ 6 *re(x)
y1 = ----------- - -------------
          3              3      
$$y_{1} = - \frac{\sqrt{6} i \operatorname{re}{\left(x\right)}}{3} + \frac{\sqrt{6} \operatorname{im}{\left(x\right)}}{3}$$
         ___             ___      
       \/ 6 *im(x)   I*\/ 6 *re(x)
y2 = - ----------- + -------------
            3              3      
$$y_{2} = \frac{\sqrt{6} i \operatorname{re}{\left(x\right)}}{3} - \frac{\sqrt{6} \operatorname{im}{\left(x\right)}}{3}$$
y2 = sqrt(6)*i*re(x)/3 - sqrt(6)*im(x)/3
Sum and product of roots [src]
sum
  ___             ___             ___             ___      
\/ 6 *im(x)   I*\/ 6 *re(x)     \/ 6 *im(x)   I*\/ 6 *re(x)
----------- - ------------- + - ----------- + -------------
     3              3                3              3      
$$\left(- \frac{\sqrt{6} i \operatorname{re}{\left(x\right)}}{3} + \frac{\sqrt{6} \operatorname{im}{\left(x\right)}}{3}\right) + \left(\frac{\sqrt{6} i \operatorname{re}{\left(x\right)}}{3} - \frac{\sqrt{6} \operatorname{im}{\left(x\right)}}{3}\right)$$
=
0
$$0$$
product
/  ___             ___      \ /    ___             ___      \
|\/ 6 *im(x)   I*\/ 6 *re(x)| |  \/ 6 *im(x)   I*\/ 6 *re(x)|
|----------- - -------------|*|- ----------- + -------------|
\     3              3      / \       3              3      /
$$\left(- \frac{\sqrt{6} i \operatorname{re}{\left(x\right)}}{3} + \frac{\sqrt{6} \operatorname{im}{\left(x\right)}}{3}\right) \left(\frac{\sqrt{6} i \operatorname{re}{\left(x\right)}}{3} - \frac{\sqrt{6} \operatorname{im}{\left(x\right)}}{3}\right)$$
=
                     2
-2*(-im(x) + I*re(x)) 
----------------------
          3           
$$- \frac{2 \left(i \operatorname{re}{\left(x\right)} - \operatorname{im}{\left(x\right)}\right)^{2}}{3}$$
-2*(-im(x) + i*re(x))^2/3