Sr Examen

Lang:

Otras calculadoras

¿Cómo usar?
La ecuación:
Ecuación x^2+3x=4
Ecuación -24-y=-16
Ecuación (11x+14)-(5x-8)=25
Ecuación 4(x-6)=x-9
Expresar {x} en función de y en la ecuación:
-18*x+8*y=-2
15*x+1*y=19
-3*x+17*y=-4
-14*x-12*y=5
Expresiones idénticas
atan(y/ tres)/ tres =Const+x^ dos / dos
arco tangente de gente de (y dividir por 3) dividir por 3 es igual a Const más x al cuadrado dividir por 2
arco tangente de gente de (y dividir por tres) dividir por tres es igual a Const más x en el grado dos dividir por dos
atan(y/3)/3=Const+x2/2
atany/3/3=Const+x2/2
atan(y/3)/3=Const+x²/2
atan(y/3)/3=Const+x en el grado 2/2
atany/3/3=Const+x^2/2
atan(y dividir por 3) dividir por 3=Const+x^2 dividir por 2
Expresiones semejantes
atan(y/3)/3=Const-x^2/2
arctan(y/3)/3=Const+x^2/2
Expresiones con funciones
Arcotangente arctan
atan(x)-(1/x^3)=0
atan(x)+x-1=0
atan(x)=(2*(-1)^(m*m)*x^2+(-1)^((m*x)^2)+m^2+5*m-x^2+4)/4
atan(x)+atan(3*x)=atan(4*x/(1-3*x^2))
atan(x/3)=0

atan(y/3)/3=Const+x^2/2 la ecuación

El profesor se sorprenderá mucho al ver tu solución correcta😉

Solución

Ha introducido [src]

    /y\         
atan|-|        2
    \3/       x 
------- = c + --
   3          2

\frac{\operatorname{atan}{\left(\frac{y}{3} \right)}}{3} = c + \frac{x^{2}}{2}

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

\frac{\operatorname{atan}{\left(\frac{y}{3} \right)}}{3} = c + \frac{x^{2}}{2}

\left(- c - \frac{x^{2}}{2}\right) + \frac{\operatorname{atan}{\left(\frac{y}{3} \right)}}{3} = 0

Abramos la expresión en la ecuación

\left(- c - \frac{x^{2}}{2}\right) + \frac{\operatorname{atan}{\left(\frac{y}{3} \right)}}{3} = 0

Obtenemos la ecuación cuadrática

- c - \frac{x^{2}}{2} + \frac{\operatorname{atan}{\left(\frac{y}{3} \right)}}{3} = 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 = - \frac{1}{2}

b = 0

c = - c + \frac{\operatorname{atan}{\left(\frac{y}{3} \right)}}{3}

, entonces

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

(0)^2 - 4 * (-1/2) * (-c + atan(y/3)/3) = -2*c + 2*atan(y/3)/3

La ecuación tiene dos raíces.

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

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

x_{1} = - \sqrt{- 2 c + \frac{2 \operatorname{atan}{\left(\frac{y}{3} \right)}}{3}}

x_{2} = \sqrt{- 2 c + \frac{2 \operatorname{atan}{\left(\frac{y}{3} \right)}}{3}}

Teorema de Cardano-Vieta

reescribamos la ecuación

\frac{\operatorname{atan}{\left(\frac{y}{3} \right)}}{3} = c + \frac{x^{2}}{2}

a x^{2} + b x + c = 0

como ecuación cuadrática reducida

x^{2} + \frac{b x}{a} + \frac{c}{a} = 0

2 c + x^{2} - \frac{2 \operatorname{atan}{\left(\frac{y}{3} \right)}}{3} = 0

p x + q + x^{2} = 0

donde

p = \frac{b}{a}

p = 0

q = \frac{c}{a}

q = 2 c - \frac{2 \operatorname{atan}{\left(\frac{y}{3} \right)}}{3}

Fórmulas de Cardano-Vieta

x_{1} + x_{2} = - p

x_{1} x_{2} = q

x_{1} + x_{2} = 0

x_{1} x_{2} = 2 c - \frac{2 \operatorname{atan}{\left(\frac{y}{3} \right)}}{3}

Gráfica

Respuesta rápida [src]

            _____________________________________________________________    /     /                /    /y\\                  /    /y\\\\          _____________________________________________________________    /     /                /    /y\\                  /    /y\\\\
           /                            2                              2     |atan2|-18*im(c) + 6*im|atan|-||, -18*re(c) + 6*re|atan|-||||         /                            2                              2     |atan2|-18*im(c) + 6*im|atan|-||, -18*re(c) + 6*re|atan|-||||
          /  /                /    /y\\\    /                /    /y\\\      |     \                \    \3//                  \    \3///|        /  /                /    /y\\\    /                /    /y\\\      |     \                \    \3//                  \    \3///|
       4 /   |-18*im(c) + 6*im|atan|-|||  + |-18*re(c) + 6*re|atan|-|||  *cos|-----------------------------------------------------------|   I*4 /   |-18*im(c) + 6*im|atan|-|||  + |-18*re(c) + 6*re|atan|-|||  *sin|-----------------------------------------------------------|
       \/    \                \    \3///    \                \    \3///      \                             2                             /     \/    \                \    \3///    \                \    \3///      \                             2                             /
x1 = - ----------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------
                                                                        3                                                                                                                                      3

x_{1} = - \frac{i \sqrt[4]{\left(- 18 \operatorname{re}{\left(c\right)} + 6 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)}\right)^{2} + \left(- 18 \operatorname{im}{\left(c\right)} + 6 \operatorname{im}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 18 \operatorname{im}{\left(c\right)} + 6 \operatorname{im}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)},- 18 \operatorname{re}{\left(c\right)} + 6 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)} \right)}}{2} \right)}}{3} - \frac{\sqrt[4]{\left(- 18 \operatorname{re}{\left(c\right)} + 6 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)}\right)^{2} + \left(- 18 \operatorname{im}{\left(c\right)} + 6 \operatorname{im}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 18 \operatorname{im}{\left(c\right)} + 6 \operatorname{im}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)},- 18 \operatorname{re}{\left(c\right)} + 6 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)} \right)}}{2} \right)}}{3}

          _____________________________________________________________    /     /                /    /y\\                  /    /y\\\\          _____________________________________________________________    /     /                /    /y\\                  /    /y\\\\
         /                            2                              2     |atan2|-18*im(c) + 6*im|atan|-||, -18*re(c) + 6*re|atan|-||||         /                            2                              2     |atan2|-18*im(c) + 6*im|atan|-||, -18*re(c) + 6*re|atan|-||||
        /  /                /    /y\\\    /                /    /y\\\      |     \                \    \3//                  \    \3///|        /  /                /    /y\\\    /                /    /y\\\      |     \                \    \3//                  \    \3///|
     4 /   |-18*im(c) + 6*im|atan|-|||  + |-18*re(c) + 6*re|atan|-|||  *cos|-----------------------------------------------------------|   I*4 /   |-18*im(c) + 6*im|atan|-|||  + |-18*re(c) + 6*re|atan|-|||  *sin|-----------------------------------------------------------|
     \/    \                \    \3///    \                \    \3///      \                             2                             /     \/    \                \    \3///    \                \    \3///      \                             2                             /
x2 = ----------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------
                                                                      3                                                                                                                                      3

x_{2} = \frac{i \sqrt[4]{\left(- 18 \operatorname{re}{\left(c\right)} + 6 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)}\right)^{2} + \left(- 18 \operatorname{im}{\left(c\right)} + 6 \operatorname{im}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 18 \operatorname{im}{\left(c\right)} + 6 \operatorname{im}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)},- 18 \operatorname{re}{\left(c\right)} + 6 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)} \right)}}{2} \right)}}{3} + \frac{\sqrt[4]{\left(- 18 \operatorname{re}{\left(c\right)} + 6 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)}\right)^{2} + \left(- 18 \operatorname{im}{\left(c\right)} + 6 \operatorname{im}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 18 \operatorname{im}{\left(c\right)} + 6 \operatorname{im}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)},- 18 \operatorname{re}{\left(c\right)} + 6 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)} \right)}}{2} \right)}}{3}

x2 = i*((-18*re(c) + 6*re(atan(y/3)))^2 + (-18*im(c) + 6*im(atan(y/3)))^2)^(1/4)*sin(atan2(-18*im(c) + 6*im(atan(y/3), -18*re(c) + 6*re(atan(y/3)))/2)/3 + ((-18*re(c) + 6*re(atan(y/3)))^2 + (-18*im(c) + 6*im(atan(y/3)))^2)^(1/4)*cos(atan2(-18*im(c) + 6*im(atan(y/3)), -18*re(c) + 6*re(atan(y/3)))/2)/3)

Suma y producto de raíces [src]

suma

       _____________________________________________________________    /     /                /    /y\\                  /    /y\\\\          _____________________________________________________________    /     /                /    /y\\                  /    /y\\\\        _____________________________________________________________    /     /                /    /y\\                  /    /y\\\\          _____________________________________________________________    /     /                /    /y\\                  /    /y\\\\
      /                            2                              2     |atan2|-18*im(c) + 6*im|atan|-||, -18*re(c) + 6*re|atan|-||||         /                            2                              2     |atan2|-18*im(c) + 6*im|atan|-||, -18*re(c) + 6*re|atan|-||||       /                            2                              2     |atan2|-18*im(c) + 6*im|atan|-||, -18*re(c) + 6*re|atan|-||||         /                            2                              2     |atan2|-18*im(c) + 6*im|atan|-||, -18*re(c) + 6*re|atan|-||||
     /  /                /    /y\\\    /                /    /y\\\      |     \                \    \3//                  \    \3///|        /  /                /    /y\\\    /                /    /y\\\      |     \                \    \3//                  \    \3///|      /  /                /    /y\\\    /                /    /y\\\      |     \                \    \3//                  \    \3///|        /  /                /    /y\\\    /                /    /y\\\      |     \                \    \3//                  \    \3///|
  4 /   |-18*im(c) + 6*im|atan|-|||  + |-18*re(c) + 6*re|atan|-|||  *cos|-----------------------------------------------------------|   I*4 /   |-18*im(c) + 6*im|atan|-|||  + |-18*re(c) + 6*re|atan|-|||  *sin|-----------------------------------------------------------|   4 /   |-18*im(c) + 6*im|atan|-|||  + |-18*re(c) + 6*re|atan|-|||  *cos|-----------------------------------------------------------|   I*4 /   |-18*im(c) + 6*im|atan|-|||  + |-18*re(c) + 6*re|atan|-|||  *sin|-----------------------------------------------------------|
  \/    \                \    \3///    \                \    \3///      \                             2                             /     \/    \                \    \3///    \                \    \3///      \                             2                             /   \/    \                \    \3///    \                \    \3///      \                             2                             /     \/    \                \    \3///    \                \    \3///      \                             2                             /
- ----------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------
                                                                   3                                                                                                                                      3                                                                                                                                      3                                                                                                                                      3

\left(- \frac{i \sqrt[4]{\left(- 18 \operatorname{re}{\left(c\right)} + 6 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)}\right)^{2} + \left(- 18 \operatorname{im}{\left(c\right)} + 6 \operatorname{im}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 18 \operatorname{im}{\left(c\right)} + 6 \operatorname{im}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)},- 18 \operatorname{re}{\left(c\right)} + 6 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)} \right)}}{2} \right)}}{3} - \frac{\sqrt[4]{\left(- 18 \operatorname{re}{\left(c\right)} + 6 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)}\right)^{2} + \left(- 18 \operatorname{im}{\left(c\right)} + 6 \operatorname{im}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 18 \operatorname{im}{\left(c\right)} + 6 \operatorname{im}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)},- 18 \operatorname{re}{\left(c\right)} + 6 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)} \right)}}{2} \right)}}{3}\right) + \left(\frac{i \sqrt[4]{\left(- 18 \operatorname{re}{\left(c\right)} + 6 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)}\right)^{2} + \left(- 18 \operatorname{im}{\left(c\right)} + 6 \operatorname{im}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 18 \operatorname{im}{\left(c\right)} + 6 \operatorname{im}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)},- 18 \operatorname{re}{\left(c\right)} + 6 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)} \right)}}{2} \right)}}{3} + \frac{\sqrt[4]{\left(- 18 \operatorname{re}{\left(c\right)} + 6 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)}\right)^{2} + \left(- 18 \operatorname{im}{\left(c\right)} + 6 \operatorname{im}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 18 \operatorname{im}{\left(c\right)} + 6 \operatorname{im}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)},- 18 \operatorname{re}{\left(c\right)} + 6 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)} \right)}}{2} \right)}}{3}\right)

0

producto

/       _____________________________________________________________    /     /                /    /y\\                  /    /y\\\\          _____________________________________________________________    /     /                /    /y\\                  /    /y\\\\\ /     _____________________________________________________________    /     /                /    /y\\                  /    /y\\\\          _____________________________________________________________    /     /                /    /y\\                  /    /y\\\\\
|      /                            2                              2     |atan2|-18*im(c) + 6*im|atan|-||, -18*re(c) + 6*re|atan|-||||         /                            2                              2     |atan2|-18*im(c) + 6*im|atan|-||, -18*re(c) + 6*re|atan|-||||| |    /                            2                              2     |atan2|-18*im(c) + 6*im|atan|-||, -18*re(c) + 6*re|atan|-||||         /                            2                              2     |atan2|-18*im(c) + 6*im|atan|-||, -18*re(c) + 6*re|atan|-|||||
|     /  /                /    /y\\\    /                /    /y\\\      |     \                \    \3//                  \    \3///|        /  /                /    /y\\\    /                /    /y\\\      |     \                \    \3//                  \    \3///|| |   /  /                /    /y\\\    /                /    /y\\\      |     \                \    \3//                  \    \3///|        /  /                /    /y\\\    /                /    /y\\\      |     \                \    \3//                  \    \3///||
|  4 /   |-18*im(c) + 6*im|atan|-|||  + |-18*re(c) + 6*re|atan|-|||  *cos|-----------------------------------------------------------|   I*4 /   |-18*im(c) + 6*im|atan|-|||  + |-18*re(c) + 6*re|atan|-|||  *sin|-----------------------------------------------------------|| |4 /   |-18*im(c) + 6*im|atan|-|||  + |-18*re(c) + 6*re|atan|-|||  *cos|-----------------------------------------------------------|   I*4 /   |-18*im(c) + 6*im|atan|-|||  + |-18*re(c) + 6*re|atan|-|||  *sin|-----------------------------------------------------------||
|  \/    \                \    \3///    \                \    \3///      \                             2                             /     \/    \                \    \3///    \                \    \3///      \                             2                             /| |\/    \                \    \3///    \                \    \3///      \                             2                             /     \/    \                \    \3///    \                \    \3///      \                             2                             /|
|- ----------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------------------------------------------------------------------------------------------|*|----------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------|
\                                                                   3                                                                                                                                      3                                                                  / \                                                                 3                                                                                                                                      3                                                                  /

\left(- \frac{i \sqrt[4]{\left(- 18 \operatorname{re}{\left(c\right)} + 6 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)}\right)^{2} + \left(- 18 \operatorname{im}{\left(c\right)} + 6 \operatorname{im}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 18 \operatorname{im}{\left(c\right)} + 6 \operatorname{im}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)},- 18 \operatorname{re}{\left(c\right)} + 6 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)} \right)}}{2} \right)}}{3} - \frac{\sqrt[4]{\left(- 18 \operatorname{re}{\left(c\right)} + 6 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)}\right)^{2} + \left(- 18 \operatorname{im}{\left(c\right)} + 6 \operatorname{im}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 18 \operatorname{im}{\left(c\right)} + 6 \operatorname{im}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)},- 18 \operatorname{re}{\left(c\right)} + 6 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)} \right)}}{2} \right)}}{3}\right) \left(\frac{i \sqrt[4]{\left(- 18 \operatorname{re}{\left(c\right)} + 6 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)}\right)^{2} + \left(- 18 \operatorname{im}{\left(c\right)} + 6 \operatorname{im}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 18 \operatorname{im}{\left(c\right)} + 6 \operatorname{im}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)},- 18 \operatorname{re}{\left(c\right)} + 6 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)} \right)}}{2} \right)}}{3} + \frac{\sqrt[4]{\left(- 18 \operatorname{re}{\left(c\right)} + 6 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)}\right)^{2} + \left(- 18 \operatorname{im}{\left(c\right)} + 6 \operatorname{im}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 18 \operatorname{im}{\left(c\right)} + 6 \operatorname{im}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)},- 18 \operatorname{re}{\left(c\right)} + 6 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)} \right)}}{2} \right)}}{3}\right)

        _________________________________________________________         /                /    /y\\                  /    /y\\\
       /                          2                            2   I*atan2|-18*im(c) + 6*im|atan|-||, -18*re(c) + 6*re|atan|-|||
      /  /    /    /y\\          \    /    /    /y\\          \           \                \    \3//                  \    \3///
-2*  /   |- im|atan|-|| + 3*im(c)|  + |- re|atan|-|| + 3*re(c)|  *e                                                             
   \/    \    \    \3//          /    \    \    \3//          /                                                                 
--------------------------------------------------------------------------------------------------------------------------------
                                                               3

- \frac{2 \sqrt{\left(3 \operatorname{re}{\left(c\right)} - \operatorname{re}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)}\right)^{2} + \left(3 \operatorname{im}{\left(c\right)} - \operatorname{im}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)}\right)^{2}} e^{i \operatorname{atan_{2}}{\left(- 18 \operatorname{im}{\left(c\right)} + 6 \operatorname{im}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)},- 18 \operatorname{re}{\left(c\right)} + 6 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{y}{3} \right)}\right)} \right)}}}{3}

-2*sqrt((-im(atan(y/3)) + 3*im(c))^2 + (-re(atan(y/3)) + 3*re(c))^2)*exp(i*atan2(-18*im(c) + 6*im(atan(y/3)), -18*re(c) + 6*re(atan(y/3))))/3

Otras calculadoras

¿Cómo usar?

Expresiones idénticas

Expresiones semejantes

Expresiones con funciones

atan(y/3)/3=Const+x^2/2 la ecuación

Solución numérica:

Solución