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(x/2)^2+y^2=25 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
   2          
/x\     2     
|-|  + y  = 25
\2/
y2+(x2)2=25y^{2} + \left(\frac{x}{2}\right)^{2} = 25
Simplificar
Solución detallada
Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.

La ecuación se convierte de
y2+(x2)2=25y^{2} + \left(\frac{x}{2}\right)^{2} = 25
en
(y2+(x2)2)−25=0\left(y^{2} + \left(\frac{x}{2}\right)^{2}\right) - 25 = 0
Abramos la expresión en la ecuación
(y2+(x2)2)−25=0\left(y^{2} + \left(\frac{x}{2}\right)^{2}\right) - 25 = 0
Obtenemos la ecuación cuadrática
x24+y2−25=0\frac{x^{2}}{4} + y^{2} - 25 = 0
Es la ecuación de la forma
a*x^2 + b*x + c = 0

La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
x1=D−b2ax_{1} = \frac{\sqrt{D} - b}{2 a}
x2=−D−b2ax_{2} = \frac{- \sqrt{D} - b}{2 a}
donde D = b^2 - 4*a*c es el discriminante.
Como
a=14a = \frac{1}{4}
b=0b = 0
c=y2−25c = y^{2} - 25
, entonces
D = b^2 - 4 * a * c = 

(0)^2 - 4 * (1/4) * (-25 + y^2) = 25 - y^2

La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)

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

o
x1=225−y2x_{1} = 2 \sqrt{25 - y^{2}}
x2=−225−y2x_{2} = - 2 \sqrt{25 - y^{2}}
Teorema de Cardano-Vieta
reescribamos la ecuación
y2+(x2)2=25y^{2} + \left(\frac{x}{2}\right)^{2} = 25
de
ax2+bx+c=0a x^{2} + b x + c = 0
como ecuación cuadrática reducida
x2+bxa+ca=0x^{2} + \frac{b x}{a} + \frac{c}{a} = 0
x2+4y2−100=0x^{2} + 4 y^{2} - 100 = 0
px+q+x2=0p x + q + x^{2} = 0
donde
p=bap = \frac{b}{a}
p=0p = 0
q=caq = \frac{c}{a}
q=4y2−100q = 4 y^{2} - 100
Fórmulas de Cardano-Vieta
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=0x_{1} + x_{2} = 0
x1x2=4y2−100x_{1} x_{2} = 4 y^{2} - 100
Gráfica
Suma y producto de raíces [src] 
suma
        ___________________________________________                                                            ___________________________________________                                                          ___________________________________________                                                            ___________________________________________                                                 
       /                       2                       /     /                       2        2   \\          /                       2                       /     /                       2        2   \\        /                       2                       /     /                       2        2   \\          /                       2                       /     /                       2        2   \\
    4 /  /       2        2   \        2      2        |atan2\-2*im(y)*re(y), 25 + im (y) - re (y)/|       4 /  /       2        2   \        2      2        |atan2\-2*im(y)*re(y), 25 + im (y) - re (y)/|     4 /  /       2        2   \        2      2        |atan2\-2*im(y)*re(y), 25 + im (y) - re (y)/|       4 /  /       2        2   \        2      2        |atan2\-2*im(y)*re(y), 25 + im (y) - re (y)/|
- 2*\/   \25 + im (y) - re (y)/  + 4*im (y)*re (y) *cos|-------------------------------------------| - 2*I*\/   \25 + im (y) - re (y)/  + 4*im (y)*re (y) *sin|-------------------------------------------| + 2*\/   \25 + im (y) - re (y)/  + 4*im (y)*re (y) *cos|-------------------------------------------| + 2*I*\/   \25 + im (y) - re (y)/  + 4*im (y)*re (y) *sin|-------------------------------------------|
                                                       \                     2                     /                                                          \                     2                     /                                                        \                     2                     /                                                          \                     2                     /
(−2i(−(re⁡(y))2+(im⁡(y))2+25)2+4(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(−2re⁡(y)im⁡(y),−(re⁡(y))2+(im⁡(y))2+25)2)−2(−(re⁡(y))2+(im⁡(y))2+25)2+4(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(−2re⁡(y)im⁡(y),−(re⁡(y))2+(im⁡(y))2+25)2))+(2i(−(re⁡(y))2+(im⁡(y))2+25)2+4(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(−2re⁡(y)im⁡(y),−(re⁡(y))2+(im⁡(y))2+25)2)+2(−(re⁡(y))2+(im⁡(y))2+25)2+4(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(−2re⁡(y)im⁡(y),−(re⁡(y))2+(im⁡(y))2+25)2))\left(- 2 i \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 25\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} + 25 \right)}}{2} \right)} - 2 \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 25\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} + 25 \right)}}{2} \right)}\right) + \left(2 i \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 25\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} + 25 \right)}}{2} \right)} + 2 \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 25\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} + 25 \right)}}{2} \right)}\right) 
=
0
00 
producto
/        ___________________________________________                                                            ___________________________________________                                                 \ /      ___________________________________________                                                            ___________________________________________                                                 \
|       /                       2                       /     /                       2        2   \\          /                       2                       /     /                       2        2   \\| |     /                       2                       /     /                       2        2   \\          /                       2                       /     /                       2        2   \\|
|    4 /  /       2        2   \        2      2        |atan2\-2*im(y)*re(y), 25 + im (y) - re (y)/|       4 /  /       2        2   \        2      2        |atan2\-2*im(y)*re(y), 25 + im (y) - re (y)/|| |  4 /  /       2        2   \        2      2        |atan2\-2*im(y)*re(y), 25 + im (y) - re (y)/|       4 /  /       2        2   \        2      2        |atan2\-2*im(y)*re(y), 25 + im (y) - re (y)/||
|- 2*\/   \25 + im (y) - re (y)/  + 4*im (y)*re (y) *cos|-------------------------------------------| - 2*I*\/   \25 + im (y) - re (y)/  + 4*im (y)*re (y) *sin|-------------------------------------------||*|2*\/   \25 + im (y) - re (y)/  + 4*im (y)*re (y) *cos|-------------------------------------------| + 2*I*\/   \25 + im (y) - re (y)/  + 4*im (y)*re (y) *sin|-------------------------------------------||
\                                                       \                     2                     /                                                          \                     2                     // \                                                     \                     2                     /                                                          \                     2                     //
(−2i(−(re⁡(y))2+(im⁡(y))2+25)2+4(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(−2re⁡(y)im⁡(y),−(re⁡(y))2+(im⁡(y))2+25)2)−2(−(re⁡(y))2+(im⁡(y))2+25)2+4(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(−2re⁡(y)im⁡(y),−(re⁡(y))2+(im⁡(y))2+25)2))(2i(−(re⁡(y))2+(im⁡(y))2+25)2+4(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(−2re⁡(y)im⁡(y),−(re⁡(y))2+(im⁡(y))2+25)2)+2(−(re⁡(y))2+(im⁡(y))2+25)2+4(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(−2re⁡(y)im⁡(y),−(re⁡(y))2+(im⁡(y))2+25)2))\left(- 2 i \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 25\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} + 25 \right)}}{2} \right)} - 2 \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 25\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} + 25 \right)}}{2} \right)}\right) \left(2 i \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 25\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} + 25 \right)}}{2} \right)} + 2 \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 25\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} + 25 \right)}}{2} \right)}\right) 
=
       ___________________________________________                                               
      /                       2                            /                       2        2   \
     /  /       2        2   \        2      2      I*atan2\-2*im(y)*re(y), 25 + im (y) - re (y)/
-4*\/   \25 + im (y) - re (y)/  + 4*im (y)*re (y) *e
−4(−(re⁡(y))2+(im⁡(y))2+25)2+4(re⁡(y))2(im⁡(y))2eiatan2⁡(−2re⁡(y)im⁡(y),−(re⁡(y))2+(im⁡(y))2+25)- 4 \sqrt{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 25\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} + 25 \right)}} 
-4*sqrt((25 + im(y)^2 - re(y)^2)^2 + 4*im(y)^2*re(y)^2)*exp(i*atan2(-2*im(y)*re(y), 25 + im(y)^2 - re(y)^2))
Respuesta rápida [src] 
             ___________________________________________                                                            ___________________________________________                                                 
            /                       2                       /     /                       2        2   \\          /                       2                       /     /                       2        2   \\
         4 /  /       2        2   \        2      2        |atan2\-2*im(y)*re(y), 25 + im (y) - re (y)/|       4 /  /       2        2   \        2      2        |atan2\-2*im(y)*re(y), 25 + im (y) - re (y)/|
x1 = - 2*\/   \25 + im (y) - re (y)/  + 4*im (y)*re (y) *cos|-------------------------------------------| - 2*I*\/   \25 + im (y) - re (y)/  + 4*im (y)*re (y) *sin|-------------------------------------------|
                                                            \                     2                     /                                                          \                     2                     /
x1=−2i(−(re⁡(y))2+(im⁡(y))2+25)2+4(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(−2re⁡(y)im⁡(y),−(re⁡(y))2+(im⁡(y))2+25)2)−2(−(re⁡(y))2+(im⁡(y))2+25)2+4(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(−2re⁡(y)im⁡(y),−(re⁡(y))2+(im⁡(y))2+25)2)x_{1} = - 2 i \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 25\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} + 25 \right)}}{2} \right)} - 2 \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 25\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} + 25 \right)}}{2} \right)} 
           ___________________________________________                                                            ___________________________________________                                                 
          /                       2                       /     /                       2        2   \\          /                       2                       /     /                       2        2   \\
       4 /  /       2        2   \        2      2        |atan2\-2*im(y)*re(y), 25 + im (y) - re (y)/|       4 /  /       2        2   \        2      2        |atan2\-2*im(y)*re(y), 25 + im (y) - re (y)/|
x2 = 2*\/   \25 + im (y) - re (y)/  + 4*im (y)*re (y) *cos|-------------------------------------------| + 2*I*\/   \25 + im (y) - re (y)/  + 4*im (y)*re (y) *sin|-------------------------------------------|
                                                          \                     2                     /                                                          \                     2                     /
x2=2i(−(re⁡(y))2+(im⁡(y))2+25)2+4(re⁡(y))2(im⁡(y))24sin⁡(atan2⁡(−2re⁡(y)im⁡(y),−(re⁡(y))2+(im⁡(y))2+25)2)+2(−(re⁡(y))2+(im⁡(y))2+25)2+4(re⁡(y))2(im⁡(y))24cos⁡(atan2⁡(−2re⁡(y)im⁡(y),−(re⁡(y))2+(im⁡(y))2+25)2)x_{2} = 2 i \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 25\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} + 25 \right)}}{2} \right)} + 2 \sqrt[4]{\left(- \left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + 25\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} + 25 \right)}}{2} \right)} 
x2 = 2*i*((-re(y)^2 + im(y)^2 + 25)^2 + 4*re(y)^2*im(y)^2)^(1/4)*sin(atan2(-2*re(y)*im(y, -re(y)^2 + im(y)^2 + 25)/2) + 2*((-re(y)^2 + im(y)^2 + 25)^2 + 4*re(y)^2*im(y)^2)^(1/4)*cos(atan2(-2*re(y)*im(y), -re(y)^2 + im(y)^2 + 25)/2))
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