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La ecuación:
Ecuación (3x−36)⋅(x+8)=0
Ecuación x+y-4=0
Ecuación 4-x/7=x/9
Ecuación z^4=1
Expresar {x} en función de y en la ecuación:
4*x-1*y=-3
9*x+17*y=13
-2*x-19*y=3
6*x+17*y=-11
Expresiones idénticas
(cuatro *x^ tres)+(dos *x)+ uno = cero
(4 multiplicar por x al cubo ) más (2 multiplicar por x) más 1 es igual a 0
(cuatro multiplicar por x en el grado tres) más (dos multiplicar por x) más uno es igual a cero
(4*x3)+(2*x)+1=0
4*x3+2*x+1=0
(4*x³)+(2*x)+1=0
(4*x en el grado 3)+(2*x)+1=0
(4x^3)+(2x)+1=0
(4x3)+(2x)+1=0
4x3+2x+1=0
4x^3+2x+1=0
(4*x^3)+(2*x)+1=O
Expresiones semejantes
(4*x^3)-(2*x)+1=0
(4*x^3)+(2*x)-1=0

(4x^3)+(2x)+1=0 la ecuación

El profesor se sorprenderá mucho al ver tu solución correcta😉

Solución

Ha introducido [src]

   3              
4*x  + 2*x + 1 = 0

\left(4 x^{3} + 2 x\right) + 1 = 0

Simplificar

Teorema de Cardano-Vieta

reescribamos la ecuación

\left(4 x^{3} + 2 x\right) + 1 = 0

a x^{3} + b x^{2} + c x + d = 0

como ecuación cúbica reducida

x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0

x^{3} + \frac{x}{2} + \frac{1}{4} = 0

p x^{2} + q x + v + x^{3} = 0

donde

p = \frac{b}{a}

p = 0

q = \frac{c}{a}

q = \frac{1}{2}

v = \frac{d}{a}

v = \frac{1}{4}

Fórmulas de Cardano-Vieta

x_{1} + x_{2} + x_{3} = - p

x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q

x_{1} x_{2} x_{3} = v

x_{1} + x_{2} + x_{3} = 0

x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = \frac{1}{2}

x_{1} x_{2} x_{3} = \frac{1}{4}

Gráfica

Trazar el gráfico f1(x)

Trazar el gráfico f2(x)

Suma y producto de raíces [src]

suma

                                 ________________     /                                     ________________\                                    ________________     /                                       ________________\                                  ________________
                                /          _____      |                                    /          _____ |                                   /          _____      |                                      /          _____ |                                 /          _____ 
                               /  27   3*\/ 105       |                            ___    /  27   3*\/ 105  |                                  /  27   3*\/ 105       |                              ___    /  27   3*\/ 105  |                                /  27   3*\/ 105  
                            3 /   -- + ---------      |           ___            \/ 3 *3 /   -- + --------- |                               3 /   -- + ---------      |             ___            \/ 3 *3 /   -- + --------- |                             3 /   -- + --------- 
             1              \/    8        8          |         \/ 3                   \/    8        8     |                1              \/    8        8          |           \/ 3                   \/    8        8     |              1              \/    8        8     
- ----------------------- + --------------------- + I*|----------------------- + ---------------------------| + - ----------------------- + --------------------- + I*|- ----------------------- - ---------------------------| + ----------------------- - ---------------------
         ________________             6               |       ________________                6             |            ________________             6               |         ________________                6             |          ________________             3          
        /          _____                              |      /          _____                               |           /          _____                              |        /          _____                               |         /          _____                         
       /  27   3*\/ 105                               |     /  27   3*\/ 105                                |          /  27   3*\/ 105                               |       /  27   3*\/ 105                                |        /  27   3*\/ 105                          
  4*3 /   -- + ---------                              |4*3 /   -- + ---------                               |     4*3 /   -- + ---------                              |  4*3 /   -- + ---------                               |   2*3 /   -- + ---------                         
    \/    8        8                                  \  \/    8        8                                   /       \/    8        8                                  \    \/    8        8                                   /     \/    8        8

\left(- \frac{\sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{3} + \frac{1}{2 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}\right) + \left(\left(- \frac{1}{4 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}} + \frac{\sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{6} - \frac{\sqrt{3}}{4 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}\right)\right) + \left(- \frac{1}{4 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}} + \frac{\sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{6} + i \left(\frac{\sqrt{3}}{4 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{6}\right)\right)\right)

  /                                       ________________\     /                                     ________________\
  |                                      /          _____ |     |                                    /          _____ |
  |                              ___    /  27   3*\/ 105  |     |                            ___    /  27   3*\/ 105  |
  |             ___            \/ 3 *3 /   -- + --------- |     |           ___            \/ 3 *3 /   -- + --------- |
  |           \/ 3                   \/    8        8     |     |         \/ 3                   \/    8        8     |
I*|- ----------------------- - ---------------------------| + I*|----------------------- + ---------------------------|
  |         ________________                6             |     |       ________________                6             |
  |        /          _____                               |     |      /          _____                               |
  |       /  27   3*\/ 105                                |     |     /  27   3*\/ 105                                |
  |  4*3 /   -- + ---------                               |     |4*3 /   -- + ---------                               |
  \    \/    8        8                                   /     \  \/    8        8                                   /

i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{6} - \frac{\sqrt{3}}{4 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}\right) + i \left(\frac{\sqrt{3}}{4 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{6}\right)

producto

/                                 ________________     /                                     ________________\\ /                                 ________________     /                                       ________________\\ /                               ________________\
|                                /          _____      |                                    /          _____ || |                                /          _____      |                                      /          _____ || |                              /          _____ |
|                               /  27   3*\/ 105       |                            ___    /  27   3*\/ 105  || |                               /  27   3*\/ 105       |                              ___    /  27   3*\/ 105  || |                             /  27   3*\/ 105  |
|                            3 /   -- + ---------      |           ___            \/ 3 *3 /   -- + --------- || |                            3 /   -- + ---------      |             ___            \/ 3 *3 /   -- + --------- || |                          3 /   -- + --------- |
|             1              \/    8        8          |         \/ 3                   \/    8        8     || |             1              \/    8        8          |           \/ 3                   \/    8        8     || |           1              \/    8        8     |
|- ----------------------- + --------------------- + I*|----------------------- + ---------------------------||*|- ----------------------- + --------------------- + I*|- ----------------------- - ---------------------------||*|----------------------- - ---------------------|
|         ________________             6               |       ________________                6             || |         ________________             6               |         ________________                6             || |       ________________             3          |
|        /          _____                              |      /          _____                               || |        /          _____                              |        /          _____                               || |      /          _____                         |
|       /  27   3*\/ 105                               |     /  27   3*\/ 105                                || |       /  27   3*\/ 105                               |       /  27   3*\/ 105                                || |     /  27   3*\/ 105                          |
|  4*3 /   -- + ---------                              |4*3 /   -- + ---------                               || |  4*3 /   -- + ---------                              |  4*3 /   -- + ---------                               || |2*3 /   -- + ---------                         |
\    \/    8        8                                  \  \/    8        8                                   // \    \/    8        8                                  \    \/    8        8                                   // \  \/    8        8                             /

\left(- \frac{1}{4 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}} + \frac{\sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{6} + i \left(\frac{\sqrt{3}}{4 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{6}\right)\right) \left(- \frac{1}{4 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}} + \frac{\sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{6} - \frac{\sqrt{3}}{4 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}\right)\right) \left(- \frac{\sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{3} + \frac{1}{2 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}\right)

-1/4

- \frac{1}{4}

-1/4

Respuesta rápida [src]

                                      ________________     /                                     ________________\
                                     /          _____      |                                    /          _____ |
                                    /  27   3*\/ 105       |                            ___    /  27   3*\/ 105  |
                                 3 /   -- + ---------      |           ___            \/ 3 *3 /   -- + --------- |
                  1              \/    8        8          |         \/ 3                   \/    8        8     |
x1 = - ----------------------- + --------------------- + I*|----------------------- + ---------------------------|
              ________________             6               |       ________________                6             |
             /          _____                              |      /          _____                               |
            /  27   3*\/ 105                               |     /  27   3*\/ 105                                |
       4*3 /   -- + ---------                              |4*3 /   -- + ---------                               |
         \/    8        8                                  \  \/    8        8                                   /

x_{1} = - \frac{1}{4 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}} + \frac{\sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{6} + i \left(\frac{\sqrt{3}}{4 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{6}\right)

                                      ________________     /                                       ________________\
                                     /          _____      |                                      /          _____ |
                                    /  27   3*\/ 105       |                              ___    /  27   3*\/ 105  |
                                 3 /   -- + ---------      |             ___            \/ 3 *3 /   -- + --------- |
                  1              \/    8        8          |           \/ 3                   \/    8        8     |
x2 = - ----------------------- + --------------------- + I*|- ----------------------- - ---------------------------|
              ________________             6               |         ________________                6             |
             /          _____                              |        /          _____                               |
            /  27   3*\/ 105                               |       /  27   3*\/ 105                                |
       4*3 /   -- + ---------                              |  4*3 /   -- + ---------                               |
         \/    8        8                                  \    \/    8        8                                   /

x_{2} = - \frac{1}{4 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}} + \frac{\sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{6} - \frac{\sqrt{3}}{4 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}\right)

                                    ________________
                                   /          _____ 
                                  /  27   3*\/ 105  
                               3 /   -- + --------- 
                1              \/    8        8     
x3 = ----------------------- - ---------------------
            ________________             3          
           /          _____                         
          /  27   3*\/ 105                          
     2*3 /   -- + ---------                         
       \/    8        8

x_{3} = - \frac{\sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}{3} + \frac{1}{2 \sqrt[3]{\frac{27}{8} + \frac{3 \sqrt{105}}{8}}}

x3 = -(27/8 + 3*sqrt(105)/8)^(1/3)/3 + 1/(2*(27/8 + 3*sqrt(105)/8)^(1/3))

Respuesta numérica [src]

x1 = -0.385458498529624

x2 = 0.192729249264812 - 0.781942255263478*i

x3 = 0.192729249264812 + 0.781942255263478*i

x3 = 0.192729249264812 + 0.781942255263478*i

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Expresiones idénticas

Expresiones semejantes

(4*x^3)+(2*x)+1=0 la ecuación

Solución numérica:

Solución

(4x^3)+(2x)+1=0 la ecuación