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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
2x2+3y2=02 x^{2} + 3 y^{2} = 0
Simplify
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:
y1=D−b2ay_{1} = \frac{\sqrt{D} - b}{2 a}
y2=−D−b2ay_{2} = \frac{- \sqrt{D} - b}{2 a}
where D = b^2 - 4*a*c - it is the discriminant.
Because
a=3a = 3
b=0b = 0
c=2x2c = 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
y1=6−x23y_{1} = \frac{\sqrt{6} \sqrt{- x^{2}}}{3}
y2=−6−x23y_{2} = - \frac{\sqrt{6} \sqrt{- x^{2}}}{3}
Vieta's Theorem
rewrite the equation
2x2+3y2=02 x^{2} + 3 y^{2} = 0
of
ay2+by+c=0a y^{2} + b y + c = 0
as reduced quadratic equation
y2+bya+ca=0y^{2} + \frac{b y}{a} + \frac{c}{a} = 0
2x23+y2=0\frac{2 x^{2}}{3} + y^{2} = 0
py+q+y2=0p y + q + y^{2} = 0
where
p=bap = \frac{b}{a}
p=0p = 0
q=caq = \frac{c}{a}
q=2x23q = \frac{2 x^{2}}{3}
Vieta Formulas
y1+y2=−py_{1} + y_{2} = - p
y1y2=qy_{1} y_{2} = q
y1+y2=0y_{1} + y_{2} = 0
y1y2=2x23y_{1} y_{2} = \frac{2 x^{2}}{3}
The graph
Rapid solution [src] 
       ___             ___      
     \/ 6 *im(x)   I*\/ 6 *re(x)
y1 = ----------- - -------------
          3              3
y1=−6ire⁡(x)3+6im⁡(x)3y_{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
y2=6ire⁡(x)3−6im⁡(x)3y_{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
(−6ire⁡(x)3+6im⁡(x)3)+(6ire⁡(x)3−6im⁡(x)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
00 
product
/  ___             ___      \ /    ___             ___      \
|\/ 6 *im(x)   I*\/ 6 *re(x)| |  \/ 6 *im(x)   I*\/ 6 *re(x)|
|----------- - -------------|*|- ----------- + -------------|
\     3              3      / \       3              3      /
(−6ire⁡(x)3+6im⁡(x)3)(6ire⁡(x)3−6im⁡(x)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
−2(ire⁡(x)−im⁡(x))23- \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
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