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Ecuación x^4+2*x^2-8=0
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Ecuación x*(x^2+8*x+16)=5*(x+4)
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Ecuación x^2+3x=4
- Expresar {x} en función de y en la ecuación:
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- Gráfico de la función y =:
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x^3-3*x-1
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Expresiones idénticas
- x^ tres - tres *x- uno = cero
- x al cubo menos 3 multiplicar por x menos 1 es igual a 0
- x en el grado tres menos tres multiplicar por x menos uno es igual a cero
- x3-3*x-1=0
- x³-3*x-1=0
- x en el grado 3-3*x-1=0
- x^3-3x-1=0
- x3-3x-1=0
- x^3-3*x-1=O
-
Expresiones semejantes
- x^3-3*x+1=0
- x^3+3*x-1=0
x^3-3*x-1=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -3$$
$$v = \frac{d}{a}$$
$$v = -1$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -3$$
$$x_{1} x_{2} x_{3} = -1$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -3$$
$$v = \frac{d}{a}$$
$$v = -1$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -3$$
$$x_{1} x_{2} x_{3} = -1$$
Respuesta rápida
[src]
/pi\ / /pi\ ___ /pi\ \ ___ /pi\
cos|--| | sin|--| \/ 3 *cos|--| | \/ 3 *sin|--|
\9 / | \9 / \9 / / 1 \| \9 / / 1 \
x1 = - ------- + I*|- ------- - ------------- + im|----------------------------------|| + ------------- + re|----------------------------------|
2 | 2 2 | _____________ || 2 | _____________ |
| | / ___ / ___\|| | / ___ / ___\|
| | / 1 I*\/ 3 | 1 I*\/ 3 ||| | / 1 I*\/ 3 | 1 I*\/ 3 ||
| |3 / - + ------- *|- - - -------||| |3 / - + ------- *|- - - -------||
\ \\/ 2 2 \ 2 2 /// \\/ 2 2 \ 2 2 //
$$x_{1} = - \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)}\right)$$
/pi\ / /pi\ ___ /pi\ \ ___ /pi\
cos|--| | sin|--| \/ 3 *cos|--| | \/ 3 *sin|--|
\9 / | \9 / \9 / / 1 \| \9 / / 1 \
x2 = - ------- + I*|- ------- + ------------- + im|----------------------------------|| - ------------- + re|----------------------------------|
2 | 2 2 | _____________ || 2 | _____________ |
| | / ___ / ___\|| | / ___ / ___\|
| | / 1 I*\/ 3 | 1 I*\/ 3 ||| | / 1 I*\/ 3 | 1 I*\/ 3 ||
| |3 / - + ------- *|- - + -------||| |3 / - + ------- *|- - + -------||
\ \\/ 2 2 \ 2 2 /// \\/ 2 2 \ 2 2 //
$$x_{2} = \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} - \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + i \left(\operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right)$$
_____________
/ ___
1 / 1 I*\/ 3
x3 = ------------------ + 3 / - + -------
_____________ \/ 2 2
/ ___
/ 1 I*\/ 3
3 / - + -------
\/ 2 2
$$x_{3} = \frac{1}{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}} + \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}$$
x3 = (1/2 + sqrt(3)*i/2)^(-1/3) + (1/2 + sqrt(3)*i/2)^(1/3)
Suma y producto de raíces
[src]
suma
/pi\ / /pi\ ___ /pi\ \ ___ /pi\ /pi\ / /pi\ ___ /pi\ \ ___ /pi\ _____________
cos|--| | sin|--| \/ 3 *cos|--| | \/ 3 *sin|--| cos|--| | sin|--| \/ 3 *cos|--| | \/ 3 *sin|--| / ___
\9 / | \9 / \9 / / 1 \| \9 / / 1 \ \9 / | \9 / \9 / / 1 \| \9 / / 1 \ 1 / 1 I*\/ 3
- ------- + I*|- ------- - ------------- + im|----------------------------------|| + ------------- + re|----------------------------------| + - ------- + I*|- ------- + ------------- + im|----------------------------------|| - ------------- + re|----------------------------------| + ------------------ + 3 / - + -------
2 | 2 2 | _____________ || 2 | _____________ | 2 | 2 2 | _____________ || 2 | _____________ | _____________ \/ 2 2
| | / ___ / ___\|| | / ___ / ___\| | | / ___ / ___\|| | / ___ / ___\| / ___
| | / 1 I*\/ 3 | 1 I*\/ 3 ||| | / 1 I*\/ 3 | 1 I*\/ 3 || | | / 1 I*\/ 3 | 1 I*\/ 3 ||| | / 1 I*\/ 3 | 1 I*\/ 3 || / 1 I*\/ 3
| |3 / - + ------- *|- - - -------||| |3 / - + ------- *|- - - -------|| | |3 / - + ------- *|- - + -------||| |3 / - + ------- *|- - + -------|| 3 / - + -------
\ \\/ 2 2 \ 2 2 /// \\/ 2 2 \ 2 2 // \ \\/ 2 2 \ 2 2 /// \\/ 2 2 \ 2 2 // \/ 2 2
$$\left(\left(\operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} - \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + i \left(\operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right)\right) + \left(- \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)}\right)\right)\right) + \left(\frac{1}{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}} + \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}\right)$$
=
_____________ / /pi\ ___ /pi\ \ / /pi\ ___ /pi\ \
/ ___ | sin|--| \/ 3 *cos|--| | | sin|--| \/ 3 *cos|--| |
1 / 1 I*\/ 3 /pi\ | \9 / \9 / / 1 \| | \9 / \9 / / 1 \| / 1 \ / 1 \
------------------ + 3 / - + ------- - cos|--| + I*|- ------- + ------------- + im|----------------------------------|| + I*|- ------- - ------------- + im|----------------------------------|| + re|----------------------------------| + re|----------------------------------|
_____________ \/ 2 2 \9 / | 2 2 | _____________ || | 2 2 | _____________ || | _____________ | | _____________ |
/ ___ | | / ___ / ___\|| | | / ___ / ___\|| | / ___ / ___\| | / ___ / ___\|
/ 1 I*\/ 3 | | / 1 I*\/ 3 | 1 I*\/ 3 ||| | | / 1 I*\/ 3 | 1 I*\/ 3 ||| | / 1 I*\/ 3 | 1 I*\/ 3 || | / 1 I*\/ 3 | 1 I*\/ 3 ||
3 / - + ------- | |3 / - + ------- *|- - + -------||| | |3 / - + ------- *|- - - -------||| |3 / - + ------- *|- - + -------|| |3 / - + ------- *|- - - -------||
\/ 2 2 \ \\/ 2 2 \ 2 2 /// \ \\/ 2 2 \ 2 2 /// \\/ 2 2 \ 2 2 // \\/ 2 2 \ 2 2 //
$$- \cos{\left(\frac{\pi}{9} \right)} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} + \frac{1}{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}} + i \left(\operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right) + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)}\right) + \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}$$
producto
/ /pi\ / /pi\ ___ /pi\ \ ___ /pi\ \ / /pi\ / /pi\ ___ /pi\ \ ___ /pi\ \ / _____________\ | cos|--| | sin|--| \/ 3 *cos|--| | \/ 3 *sin|--| | | cos|--| | sin|--| \/ 3 *cos|--| | \/ 3 *sin|--| | | / ___ | | \9 / | \9 / \9 / / 1 \| \9 / / 1 \| | \9 / | \9 / \9 / / 1 \| \9 / / 1 \| | 1 / 1 I*\/ 3 | |- ------- + I*|- ------- - ------------- + im|----------------------------------|| + ------------- + re|----------------------------------||*|- ------- + I*|- ------- + ------------- + im|----------------------------------|| - ------------- + re|----------------------------------||*|------------------ + 3 / - + ------- | | 2 | 2 2 | _____________ || 2 | _____________ || | 2 | 2 2 | _____________ || 2 | _____________ || | _____________ \/ 2 2 | | | | / ___ / ___\|| | / ___ / ___\|| | | | / ___ / ___\|| | / ___ / ___\|| | / ___ | | | | / 1 I*\/ 3 | 1 I*\/ 3 ||| | / 1 I*\/ 3 | 1 I*\/ 3 ||| | | | / 1 I*\/ 3 | 1 I*\/ 3 ||| | / 1 I*\/ 3 | 1 I*\/ 3 ||| | / 1 I*\/ 3 | | | |3 / - + ------- *|- - - -------||| |3 / - + ------- *|- - - -------||| | | |3 / - + ------- *|- - + -------||| |3 / - + ------- *|- - + -------||| |3 / - + ------- | \ \ \\/ 2 2 \ 2 2 /// \\/ 2 2 \ 2 2 /// \ \ \\/ 2 2 \ 2 2 /// \\/ 2 2 \ 2 2 /// \\/ 2 2 /
$$\left(- \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)}\right)\right) \left(\operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} - \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + i \left(\operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right)\right) \left(\frac{1}{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}} + \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}\right)$$
=
/ / 5*pi*I\ -5*pi*I\
| | ------| -------|
| | 9 | 18 | /pi\
2*\-1 + I*\1 - e /*e /*cos|--|
\9 /
$$2 \left(-1 + i \left(1 - e^{\frac{5 i \pi}{9}}\right) e^{- \frac{5 i \pi}{18}}\right) \cos{\left(\frac{\pi}{9} \right)}$$
2*(-1 + i*(1 - exp(5*pi*i/9))*exp(-5*pi*i/18))*cos(pi/9)
Respuesta numérica
[src]
x1 = 1.87938524157182
x2 = -1.53208888623796
x3 = -0.347296355333861
x3 = -0.347296355333861
Gráfico