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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
2x2+3y2=32 x^{2} + 3 y^{2} = 3
Simplify
Express using the variable y
Implicit function graph
Detail solution
Move right part of the equation to
left part with negative sign.

The equation is transformed from
2x2+3y2=32 x^{2} + 3 y^{2} = 3
to
(2x2+3y2)−3=0\left(2 x^{2} + 3 y^{2}\right) - 3 = 0
This equation is of the form
a∗x2+b∗x+c=0a*x^2 + b*x + c = 0
A quadratic equation can be solved using the discriminant
The roots of the quadratic equation:
x1=D−b2ax_{1} = \frac{\sqrt{D} - b}{2 a}
x2=−D−b2ax_{2} = \frac{- \sqrt{D} - b}{2 a}
where D=b2−4acD = b^2 - 4 a c is the discriminant.
Because
a=2a = 2
b=0b = 0
c=3y2−3c = 3 y^{2} - 3
, then
D=b2−4∗a∗c=D = b^2 - 4 * a * c =
−2⋅4⋅(3y2−3)+02=−24y2+24- 2 \cdot 4 \cdot \left(3 y^{2} - 3\right) + 0^{2} = - 24 y^{2} + 24
The equation has two roots.
x1=(−b+D)2ax_1 = \frac{(-b + \sqrt{D})}{2 a}
x2=(−b−D)2ax_2 = \frac{(-b - \sqrt{D})}{2 a}
or
x1=−24y2+244x_{1} = \frac{\sqrt{- 24 y^{2} + 24}}{4}
Simplify
x2=−−24y2+244x_{2} = - \frac{\sqrt{- 24 y^{2} + 24}}{4}
Simplify
Vieta's Theorem
rewrite the equation
2x2+3y2=32 x^{2} + 3 y^{2} = 3
of
ax2+bx+c=0a x^{2} + b x + c = 0
as reduced quadratic equation
x2+bxa+ca=0x^{2} + \frac{b x}{a} + \frac{c}{a} = 0
x2+3y22−32=0x^{2} + \frac{3 y^{2}}{2} - \frac{3}{2} = 0
px+x2+q=0p x + x^{2} + q = 0
where
p=bap = \frac{b}{a}
p=0p = 0
q=caq = \frac{c}{a}
q=3y22−32q = \frac{3 y^{2}}{2} - \frac{3}{2}
Vieta Formulas
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=0x_{1} + x_{2} = 0
x1x2=3y22−32x_{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
(−−6y2+62)+(−6y2+62)\left(- \frac{\sqrt{- 6 y^{2} + 6}}{2}\right) + \left(\frac{\sqrt{- 6 y^{2} + 6}}{2}\right) 
=
0
00
Simplify 
product
    __________       __________
   /        2       /        2 
-\/  6 - 6*y      \/  6 - 6*y  
--------------- * -------------
       2                2
(−−6y2+62)∗(−6y2+62)\left(- \frac{\sqrt{- 6 y^{2} + 6}}{2}\right) * \left(\frac{\sqrt{- 6 y^{2} + 6}}{2}\right) 
=
         2
  3   3*y 
- - + ----
  2    2
3y22−32\frac{3 y^{2}}{2} - \frac{3}{2}
Simplify
Rapid solution [src] 
          __________ 
         /        2  
      -\/  6 - 6*y   
x_1 = ---------------
             2
x1=−−6y2+62x_{1} = - \frac{\sqrt{- 6 y^{2} + 6}}{2}
Simplify 
         __________
        /        2 
      \/  6 - 6*y  
x_2 = -------------
            2
x2=−6y2+62x_{2} = \frac{\sqrt{- 6 y^{2} + 6}}{2}
Simplify
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