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x^2+y^2=4 equation

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Numerical solution:

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The solution

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 2    2    
x  + y  = 4
x2+y2=4x^{2} + y^{2} = 4
Simplify
Detail solution
Move right part of the equation to
left part with negative sign.

The equation is transformed from
x2+y2=4x^{2} + y^{2} = 4
to
(x2+y2)−4=0\left(x^{2} + y^{2}\right) - 4 = 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:
x1=D−b2ax_{1} = \frac{\sqrt{D} - b}{2 a}
x2=−D−b2ax_{2} = \frac{- \sqrt{D} - b}{2 a}
where D = b^2 - 4*a*c - it is the discriminant.
Because
a=1a = 1
b=0b = 0
c=y2−4c = y^{2} - 4
, then
D = b^2 - 4 * a * c = 

(0)^2 - 4 * (1) * (-4 + y^2) = 16 - 4*y^2

The equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

or
x1=16−4y22x_{1} = \frac{\sqrt{16 - 4 y^{2}}}{2}
x2=−16−4y22x_{2} = - \frac{\sqrt{16 - 4 y^{2}}}{2}
Vieta's Theorem
it is reduced quadratic equation
px+q+x2=0p x + q + x^{2} = 0
where
p=bap = \frac{b}{a}
p=0p = 0
q=caq = \frac{c}{a}
q=y2−4q = y^{2} - 4
Vieta Formulas
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=0x_{1} + x_{2} = 0
x1x2=y2−4x_{1} x_{2} = y^{2} - 4
The graph
Rapid solution [src] 
           __________________________________________                                                         __________________________________________                                                
          /                      2                       /     /                      2        2   \\        /                      2                       /     /                      2        2   \\
       4 /  /      2        2   \        2      2        |atan2\-2*im(y)*re(y), 4 + im (y) - re (y)/|     4 /  /      2        2   \        2      2        |atan2\-2*im(y)*re(y), 4 + im (y) - re (y)/|
x1 = - \/   \4 + im (y) - re (y)/  + 4*im (y)*re (y) *cos|------------------------------------------| - I*\/   \4 + im (y) - re (y)/  + 4*im (y)*re (y) *sin|------------------------------------------|
                                                         \                    2                     /                                                       \                    2                     /
x1=−i(−(re⁡(y))2+(im⁡(y))2+4)2+4(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(−2re⁡(y)im⁡(y),−(re⁡(y))2+(im⁡(y))2+4)2)−(−(re⁡(y))2+(im⁡(y))2+4)2+4(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(−2re⁡(y)im⁡(y),−(re⁡(y))2+(im⁡(y))2+4)2)x_{1} = - i \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4 \right)}}{2} \right)} - \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4 \right)}}{2} \right)} 
         __________________________________________                                                         __________________________________________                                                
        /                      2                       /     /                      2        2   \\        /                      2                       /     /                      2        2   \\
     4 /  /      2        2   \        2      2        |atan2\-2*im(y)*re(y), 4 + im (y) - re (y)/|     4 /  /      2        2   \        2      2        |atan2\-2*im(y)*re(y), 4 + im (y) - re (y)/|
x2 = \/   \4 + im (y) - re (y)/  + 4*im (y)*re (y) *cos|------------------------------------------| + I*\/   \4 + im (y) - re (y)/  + 4*im (y)*re (y) *sin|------------------------------------------|
                                                       \                    2                     /                                                       \                    2                     /
x2=i(−(re⁡(y))2+(im⁡(y))2+4)2+4(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(−2re⁡(y)im⁡(y),−(re⁡(y))2+(im⁡(y))2+4)2)+(−(re⁡(y))2+(im⁡(y))2+4)2+4(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(−2re⁡(y)im⁡(y),−(re⁡(y))2+(im⁡(y))2+4)2)x_{2} = i \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4 \right)}}{2} \right)} + \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4 \right)}}{2} \right)} 
x2 = i*((-re(y)^2 + im(y)^2 + 4)^2 + 4*re(y)^2*im(y)^2)^(1/4)*sin(atan2(-2*re(y)*im(y, -re(y)^2 + im(y)^2 + 4)/2) + ((-re(y)^2 + im(y)^2 + 4)^2 + 4*re(y)^2*im(y)^2)^(1/4)*cos(atan2(-2*re(y)*im(y), -re(y)^2 + im(y)^2 + 4)/2))
Sum and product of roots [src] 
sum
      __________________________________________                                                         __________________________________________                                                       __________________________________________                                                         __________________________________________                                                
     /                      2                       /     /                      2        2   \\        /                      2                       /     /                      2        2   \\      /                      2                       /     /                      2        2   \\        /                      2                       /     /                      2        2   \\
  4 /  /      2        2   \        2      2        |atan2\-2*im(y)*re(y), 4 + im (y) - re (y)/|     4 /  /      2        2   \        2      2        |atan2\-2*im(y)*re(y), 4 + im (y) - re (y)/|   4 /  /      2        2   \        2      2        |atan2\-2*im(y)*re(y), 4 + im (y) - re (y)/|     4 /  /      2        2   \        2      2        |atan2\-2*im(y)*re(y), 4 + im (y) - re (y)/|
- \/   \4 + im (y) - re (y)/  + 4*im (y)*re (y) *cos|------------------------------------------| - I*\/   \4 + im (y) - re (y)/  + 4*im (y)*re (y) *sin|------------------------------------------| + \/   \4 + im (y) - re (y)/  + 4*im (y)*re (y) *cos|------------------------------------------| + I*\/   \4 + im (y) - re (y)/  + 4*im (y)*re (y) *sin|------------------------------------------|
                                                    \                    2                     /                                                       \                    2                     /                                                     \                    2                     /                                                       \                    2                     /
(−i(−(re⁡(y))2+(im⁡(y))2+4)2+4(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(−2re⁡(y)im⁡(y),−(re⁡(y))2+(im⁡(y))2+4)2)−(−(re⁡(y))2+(im⁡(y))2+4)2+4(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(−2re⁡(y)im⁡(y),−(re⁡(y))2+(im⁡(y))2+4)2))+(i(−(re⁡(y))2+(im⁡(y))2+4)2+4(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(−2re⁡(y)im⁡(y),−(re⁡(y))2+(im⁡(y))2+4)2)+(−(re⁡(y))2+(im⁡(y))2+4)2+4(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(−2re⁡(y)im⁡(y),−(re⁡(y))2+(im⁡(y))2+4)2))\left(- i \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4 \right)}}{2} \right)} - \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4 \right)}}{2} \right)}\right) + \left(i \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4 \right)}}{2} \right)} + \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4 \right)}}{2} \right)}\right) 
=
0
00 
product
/      __________________________________________                                                         __________________________________________                                                \ /    __________________________________________                                                         __________________________________________                                                \
|     /                      2                       /     /                      2        2   \\        /                      2                       /     /                      2        2   \\| |   /                      2                       /     /                      2        2   \\        /                      2                       /     /                      2        2   \\|
|  4 /  /      2        2   \        2      2        |atan2\-2*im(y)*re(y), 4 + im (y) - re (y)/|     4 /  /      2        2   \        2      2        |atan2\-2*im(y)*re(y), 4 + im (y) - re (y)/|| |4 /  /      2        2   \        2      2        |atan2\-2*im(y)*re(y), 4 + im (y) - re (y)/|     4 /  /      2        2   \        2      2        |atan2\-2*im(y)*re(y), 4 + im (y) - re (y)/||
|- \/   \4 + im (y) - re (y)/  + 4*im (y)*re (y) *cos|------------------------------------------| - I*\/   \4 + im (y) - re (y)/  + 4*im (y)*re (y) *sin|------------------------------------------||*|\/   \4 + im (y) - re (y)/  + 4*im (y)*re (y) *cos|------------------------------------------| + I*\/   \4 + im (y) - re (y)/  + 4*im (y)*re (y) *sin|------------------------------------------||
\                                                    \                    2                     /                                                       \                    2                     // \                                                  \                    2                     /                                                       \                    2                     //
(−i(−(re⁡(y))2+(im⁡(y))2+4)2+4(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(−2re⁡(y)im⁡(y),−(re⁡(y))2+(im⁡(y))2+4)2)−(−(re⁡(y))2+(im⁡(y))2+4)2+4(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(−2re⁡(y)im⁡(y),−(re⁡(y))2+(im⁡(y))2+4)2))(i(−(re⁡(y))2+(im⁡(y))2+4)2+4(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(−2re⁡(y)im⁡(y),−(re⁡(y))2+(im⁡(y))2+4)2)+(−(re⁡(y))2+(im⁡(y))2+4)2+4(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(−2re⁡(y)im⁡(y),−(re⁡(y))2+(im⁡(y))2+4)2))\left(- i \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4 \right)}}{2} \right)} - \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4 \right)}}{2} \right)}\right) \left(i \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4 \right)}}{2} \right)} + \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4 \right)}}{2} \right)}\right) 
=
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    /                      2                            /                      2        2   \
   /  /      2        2   \        2      2      I*atan2\-2*im(y)*re(y), 4 + im (y) - re (y)/
-\/   \4 + im (y) - re (y)/  + 4*im (y)*re (y) *e
−(−(re⁡(y))2+(im⁡(y))2+4)2+4(re⁡(y))2(im⁡(y))2eiatan2⁡(−2re⁡(y)im⁡(y),−(re⁡(y))2+(im⁡(y))2+4)- \sqrt{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} e^{i \operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4 \right)}} 
-sqrt((4 + im(y)^2 - re(y)^2)^2 + 4*im(y)^2*re(y)^2)*exp(i*atan2(-2*im(y)*re(y), 4 + im(y)^2 - re(y)^2))
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