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La ecuación:
Ecuación x^2+11=0
Ecuación x^3-x^2-x-1=0
Ecuación 5x+12=8x+30
Ecuación 1/x^2-3/x-4=0
Expresar {x} en función de y en la ecuación:
-7*x-13*y=5
15*x+1*y=19
7*x-1*y=-7
20*x+5*y=-3
Expresiones idénticas
(x+y)*(x+y)+ cuatro *(x- dos *y+ cinco)= cero
(x más y) multiplicar por (x más y) más 4 multiplicar por (x menos 2 multiplicar por y más 5) es igual a 0
(x más y) multiplicar por (x más y) más cuatro multiplicar por (x menos dos multiplicar por y más cinco) es igual a cero
(x+y)(x+y)+4(x-2y+5)=0
x+yx+y+4x-2y+5=0
(x+y)*(x+y)+4*(x-2*y+5)=O
Expresiones semejantes
(x+y)*(x+y)+4*(x+2*y+5)=0
(x+y)*(x-y)+4*(x-2*y+5)=0
(x-y)*(x+y)+4*(x-2*y+5)=0
(x+y)*(x+y)+4*(x-2*y-5)=0
(x+y)*(x+y)-4*(x-2*y+5)=0

(x+y)(x+y)+4(x-2*y+5)=0 la ecuación

El profesor se sorprenderá mucho al ver tu solución correcta😉

Solución

Ha introducido [src]

(x + y)*(x + y) + 4*(x - 2*y + 5) = 0

$$\left(x + y\right) \left(x + y\right) + 4 \left(\left(x - 2 y\right) + 5\right) = 0$$

Simplificar

Solución detallada

Abramos la expresión en la ecuación
$$\left(x + y\right) \left(x + y\right) + 4 \left(\left(x - 2 y\right) + 5\right) = 0$$
Obtenemos la ecuación cuadrática
$$x^{2} + 2 x y + 4 x + y^{2} - 8 y + 20 = 0$$
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:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 1$$
$$b = 2 y + 4$$
$$c = y^{2} - 8 y + 20$$
, entonces

D = b^2 - 4 * a * c =

(4 + 2*y)^2 - 4 * (1) * (20 + y^2 - 8*y) = -80 + (4 + 2*y)^2 - 4*y^2 + 32*y

La ecuación tiene dos raíces.

x1 = (-b + sqrt(D)) / (2*a)

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

o
$$x_{1} = - y + \frac{\sqrt{- 4 y^{2} + 32 y + \left(2 y + 4\right)^{2} - 80}}{2} - 2$$
$$x_{2} = - y - \frac{\sqrt{- 4 y^{2} + 32 y + \left(2 y + 4\right)^{2} - 80}}{2} - 2$$

Gráfica

Respuesta rápida [src]

                    /              ____________________________                                  \        ____________________________                                  
                    |           4 /               2       2        /atan2(3*im(y), -4 + 3*re(y))\|     4 /               2       2        /atan2(3*im(y), -4 + 3*re(y))\
x1 = -2 - re(y) + I*|-im(y) - 2*\/  (-4 + 3*re(y))  + 9*im (y) *sin|----------------------------|| - 2*\/  (-4 + 3*re(y))  + 9*im (y) *cos|----------------------------|
                    \                                              \             2              //                                        \             2              /

$$x_{1} = i \left(- 2 \sqrt[4]{\left(3 \operatorname{re}{\left(y\right)} - 4\right)^{2} + 9 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(3 \operatorname{im}{\left(y\right)},3 \operatorname{re}{\left(y\right)} - 4 \right)}}{2} \right)} - \operatorname{im}{\left(y\right)}\right) - 2 \sqrt[4]{\left(3 \operatorname{re}{\left(y\right)} - 4\right)^{2} + 9 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(3 \operatorname{im}{\left(y\right)},3 \operatorname{re}{\left(y\right)} - 4 \right)}}{2} \right)} - \operatorname{re}{\left(y\right)} - 2$$

                    /              ____________________________                                  \        ____________________________                                  
                    |           4 /               2       2        /atan2(3*im(y), -4 + 3*re(y))\|     4 /               2       2        /atan2(3*im(y), -4 + 3*re(y))\
x2 = -2 - re(y) + I*|-im(y) + 2*\/  (-4 + 3*re(y))  + 9*im (y) *sin|----------------------------|| + 2*\/  (-4 + 3*re(y))  + 9*im (y) *cos|----------------------------|
                    \                                              \             2              //                                        \             2              /

$$x_{2} = i \left(2 \sqrt[4]{\left(3 \operatorname{re}{\left(y\right)} - 4\right)^{2} + 9 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(3 \operatorname{im}{\left(y\right)},3 \operatorname{re}{\left(y\right)} - 4 \right)}}{2} \right)} - \operatorname{im}{\left(y\right)}\right) + 2 \sqrt[4]{\left(3 \operatorname{re}{\left(y\right)} - 4\right)^{2} + 9 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(3 \operatorname{im}{\left(y\right)},3 \operatorname{re}{\left(y\right)} - 4 \right)}}{2} \right)} - \operatorname{re}{\left(y\right)} - 2$$

x2 = i*(2*((3*re(y) - 4)^2 + 9*im(y)^2)^(1/4)*sin(atan2(3*im(y, 3*re(y) - 4)/2) - im(y)) + 2*((3*re(y) - 4)^2 + 9*im(y)^2)^(1/4)*cos(atan2(3*im(y), 3*re(y) - 4)/2) - re(y) - 2)

Suma y producto de raíces [src]

suma

               /              ____________________________                                  \        ____________________________                                                    /              ____________________________                                  \        ____________________________                                  
               |           4 /               2       2        /atan2(3*im(y), -4 + 3*re(y))\|     4 /               2       2        /atan2(3*im(y), -4 + 3*re(y))\                  |           4 /               2       2        /atan2(3*im(y), -4 + 3*re(y))\|     4 /               2       2        /atan2(3*im(y), -4 + 3*re(y))\
-2 - re(y) + I*|-im(y) - 2*\/  (-4 + 3*re(y))  + 9*im (y) *sin|----------------------------|| - 2*\/  (-4 + 3*re(y))  + 9*im (y) *cos|----------------------------| + -2 - re(y) + I*|-im(y) + 2*\/  (-4 + 3*re(y))  + 9*im (y) *sin|----------------------------|| + 2*\/  (-4 + 3*re(y))  + 9*im (y) *cos|----------------------------|
               \                                              \             2              //                                        \             2              /                  \                                              \             2              //                                        \             2              /

$$\left(i \left(- 2 \sqrt[4]{\left(3 \operatorname{re}{\left(y\right)} - 4\right)^{2} + 9 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(3 \operatorname{im}{\left(y\right)},3 \operatorname{re}{\left(y\right)} - 4 \right)}}{2} \right)} - \operatorname{im}{\left(y\right)}\right) - 2 \sqrt[4]{\left(3 \operatorname{re}{\left(y\right)} - 4\right)^{2} + 9 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(3 \operatorname{im}{\left(y\right)},3 \operatorname{re}{\left(y\right)} - 4 \right)}}{2} \right)} - \operatorname{re}{\left(y\right)} - 2\right) + \left(i \left(2 \sqrt[4]{\left(3 \operatorname{re}{\left(y\right)} - 4\right)^{2} + 9 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(3 \operatorname{im}{\left(y\right)},3 \operatorname{re}{\left(y\right)} - 4 \right)}}{2} \right)} - \operatorname{im}{\left(y\right)}\right) + 2 \sqrt[4]{\left(3 \operatorname{re}{\left(y\right)} - 4\right)^{2} + 9 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(3 \operatorname{im}{\left(y\right)},3 \operatorname{re}{\left(y\right)} - 4 \right)}}{2} \right)} - \operatorname{re}{\left(y\right)} - 2\right)$$

                 /              ____________________________                                  \     /              ____________________________                                  \
                 |           4 /               2       2        /atan2(3*im(y), -4 + 3*re(y))\|     |           4 /               2       2        /atan2(3*im(y), -4 + 3*re(y))\|
-4 - 2*re(y) + I*|-im(y) - 2*\/  (-4 + 3*re(y))  + 9*im (y) *sin|----------------------------|| + I*|-im(y) + 2*\/  (-4 + 3*re(y))  + 9*im (y) *sin|----------------------------||
                 \                                              \             2              //     \                                              \             2              //

$$i \left(- 2 \sqrt[4]{\left(3 \operatorname{re}{\left(y\right)} - 4\right)^{2} + 9 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(3 \operatorname{im}{\left(y\right)},3 \operatorname{re}{\left(y\right)} - 4 \right)}}{2} \right)} - \operatorname{im}{\left(y\right)}\right) + i \left(2 \sqrt[4]{\left(3 \operatorname{re}{\left(y\right)} - 4\right)^{2} + 9 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(3 \operatorname{im}{\left(y\right)},3 \operatorname{re}{\left(y\right)} - 4 \right)}}{2} \right)} - \operatorname{im}{\left(y\right)}\right) - 2 \operatorname{re}{\left(y\right)} - 4$$

producto

/               /              ____________________________                                  \        ____________________________                                  \ /               /              ____________________________                                  \        ____________________________                                  \
|               |           4 /               2       2        /atan2(3*im(y), -4 + 3*re(y))\|     4 /               2       2        /atan2(3*im(y), -4 + 3*re(y))\| |               |           4 /               2       2        /atan2(3*im(y), -4 + 3*re(y))\|     4 /               2       2        /atan2(3*im(y), -4 + 3*re(y))\|
|-2 - re(y) + I*|-im(y) - 2*\/  (-4 + 3*re(y))  + 9*im (y) *sin|----------------------------|| - 2*\/  (-4 + 3*re(y))  + 9*im (y) *cos|----------------------------||*|-2 - re(y) + I*|-im(y) + 2*\/  (-4 + 3*re(y))  + 9*im (y) *sin|----------------------------|| + 2*\/  (-4 + 3*re(y))  + 9*im (y) *cos|----------------------------||
\               \                                              \             2              //                                        \             2              // \               \                                              \             2              //                                        \             2              //

$$\left(i \left(- 2 \sqrt[4]{\left(3 \operatorname{re}{\left(y\right)} - 4\right)^{2} + 9 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(3 \operatorname{im}{\left(y\right)},3 \operatorname{re}{\left(y\right)} - 4 \right)}}{2} \right)} - \operatorname{im}{\left(y\right)}\right) - 2 \sqrt[4]{\left(3 \operatorname{re}{\left(y\right)} - 4\right)^{2} + 9 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(3 \operatorname{im}{\left(y\right)},3 \operatorname{re}{\left(y\right)} - 4 \right)}}{2} \right)} - \operatorname{re}{\left(y\right)} - 2\right) \left(i \left(2 \sqrt[4]{\left(3 \operatorname{re}{\left(y\right)} - 4\right)^{2} + 9 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(3 \operatorname{im}{\left(y\right)},3 \operatorname{re}{\left(y\right)} - 4 \right)}}{2} \right)} - \operatorname{im}{\left(y\right)}\right) + 2 \sqrt[4]{\left(3 \operatorname{re}{\left(y\right)} - 4\right)^{2} + 9 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(3 \operatorname{im}{\left(y\right)},3 \operatorname{re}{\left(y\right)} - 4 \right)}}{2} \right)} - \operatorname{re}{\left(y\right)} - 2\right)$$

       2        2                                           
20 + re (y) - im (y) - 8*re(y) - 8*I*im(y) + 2*I*im(y)*re(y)

$$\left(\operatorname{re}{\left(y\right)}\right)^{2} + 2 i \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 8 \operatorname{re}{\left(y\right)} - \left(\operatorname{im}{\left(y\right)}\right)^{2} - 8 i \operatorname{im}{\left(y\right)} + 20$$

20 + re(y)^2 - im(y)^2 - 8*re(y) - 8*i*im(y) + 2*i*im(y)*re(y)

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Expresiones idénticas

Expresiones semejantes

(x+y)*(x+y)+4*(x-2*y+5)=0 la ecuación

Solución numérica:

Solución

(x+y)(x+y)+4(x-2*y+5)=0 la ecuación