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2x^2+3y^2=3 equation

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Numerical solution:

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The solution

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   2      2    
2*x  + 3*y  = 3
$$2 x^{2} + 3 y^{2} = 3$$
Detail solution
Move right part of the equation to
left part with negative sign.

The equation is transformed from
$$2 x^{2} + 3 y^{2} = 3$$
to
$$\left(2 x^{2} + 3 y^{2}\right) - 3 = 0$$
This equation is of the form
$$a*x^2 + b*x + c = 0$$
A quadratic equation can be solved using the discriminant
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 2$$
$$b = 0$$
$$c = 3 y^{2} - 3$$
, then
$$D = b^2 - 4 * a * c = $$
$$- 2 \cdot 4 \cdot \left(3 y^{2} - 3\right) + 0^{2} = - 24 y^{2} + 24$$
The equation has two roots.
$$x_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$x_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$x_{1} = \frac{\sqrt{- 24 y^{2} + 24}}{4}$$
Simplify
$$x_{2} = - \frac{\sqrt{- 24 y^{2} + 24}}{4}$$
Simplify
Vieta's Theorem
rewrite the equation
$$2 x^{2} + 3 y^{2} = 3$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{3 y^{2}}{2} - \frac{3}{2} = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = \frac{3 y^{2}}{2} - \frac{3}{2}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 0$$
$$x_{1} x_{2} = \frac{3 y^{2}}{2} - \frac{3}{2}$$
The graph
Sum and product of roots [src]
sum
    __________       __________
   /        2       /        2 
-\/  6 - 6*y      \/  6 - 6*y  
--------------- + -------------
       2                2      
$$\left(- \frac{\sqrt{- 6 y^{2} + 6}}{2}\right) + \left(\frac{\sqrt{- 6 y^{2} + 6}}{2}\right)$$
=
0
$$0$$
product
    __________       __________
   /        2       /        2 
-\/  6 - 6*y      \/  6 - 6*y  
--------------- * -------------
       2                2      
$$\left(- \frac{\sqrt{- 6 y^{2} + 6}}{2}\right) * \left(\frac{\sqrt{- 6 y^{2} + 6}}{2}\right)$$
=
         2
  3   3*y 
- - + ----
  2    2  
$$\frac{3 y^{2}}{2} - \frac{3}{2}$$
Rapid solution [src]
          __________ 
         /        2  
      -\/  6 - 6*y   
x_1 = ---------------
             2       
$$x_{1} = - \frac{\sqrt{- 6 y^{2} + 6}}{2}$$
         __________
        /        2 
      \/  6 - 6*y  
x_2 = -------------
            2      
$$x_{2} = \frac{\sqrt{- 6 y^{2} + 6}}{2}$$