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Ecuación cos(pi*x)=0
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Ecuación tan(x)=-5
- Ecuación xy
- Expresar {x} en función de y en la ecuación:
- -18*x-3*y=2
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- -12*x+2*y=14
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- Gráfico de la función y =:
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x^3-3x+1
- Integral de d{x}:
- x^3-3x+1
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Expresiones idénticas
- x^ tres -3x+ uno = cero
- x al cubo menos 3x más 1 es igual a 0
- x en el grado tres menos 3x más uno es igual a cero
- x3-3x+1=0
- x³-3x+1=0
- x en el grado 3-3x+1=0
- x^3-3x+1=O
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Expresiones semejantes
- x^3+3x+1=0
- x^3-3x-1=0
x^3-3x+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 / / 1 \ | \9 / / 1 \ \9 /| \9 /
x1 = ------- - 3*re|--------------------------------------| + I*|------- - 3*im|--------------------------------------| + -------------| - -------------
2 | _________________| | 2 | _________________| 2 | 2
|/ ___\ / ___ | | |/ ___\ / ___ | |
|| 1 I*\/ 3 | / 27 27*I*\/ 3 | | || 1 I*\/ 3 | / 27 27*I*\/ 3 | |
||- - - -------|*3 / -- + ---------- | | ||- - - -------|*3 / -- + ---------- | |
\\ 2 2 / \/ 2 2 / \ \\ 2 2 / \/ 2 2 / /
$$x_{1} = - \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} + i \left(- 3 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right)$$
/pi\ / /pi\ ___ /pi\\ ___ /pi\
cos|--| |sin|--| \/ 3 *cos|--|| \/ 3 *sin|--|
\9 / / 1 \ | \9 / / 1 \ \9 /| \9 /
x2 = ------- - 3*re|--------------------------------------| + I*|------- - 3*im|--------------------------------------| - -------------| + -------------
2 | _________________| | 2 | _________________| 2 | 2
|/ ___\ / ___ | | |/ ___\ / ___ | |
|| 1 I*\/ 3 | / 27 27*I*\/ 3 | | || 1 I*\/ 3 | / 27 27*I*\/ 3 | |
||- - + -------|*3 / -- + ---------- | | ||- - + -------|*3 / -- + ---------- | |
\\ 2 2 / \/ 2 2 / \ \\ 2 2 / \/ 2 2 / /
$$x_{2} = \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)}\right)$$
_________________
/ ___
/ 27 27*I*\/ 3
3 / -- + ----------
3 \/ 2 2
x3 = - ---------------------- - ----------------------
_________________ 3
/ ___
/ 27 27*I*\/ 3
3 / -- + ----------
\/ 2 2
$$x_{3} = - \frac{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}{3} - \frac{3}{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}$$
x3 = -(27/2 + 27*sqrt(3)*i/2)^(1/3)/3 - 3/(27/2 + 27*sqrt(3)*i/2)^(1/3)
Suma y producto de raíces
[src]
suma
_________________
/ ___
/pi\ / /pi\ ___ /pi\\ ___ /pi\ /pi\ / /pi\ ___ /pi\\ ___ /pi\ / 27 27*I*\/ 3
cos|--| |sin|--| \/ 3 *cos|--|| \/ 3 *sin|--| cos|--| |sin|--| \/ 3 *cos|--|| \/ 3 *sin|--| 3 / -- + ----------
\9 / / 1 \ | \9 / / 1 \ \9 /| \9 / \9 / / 1 \ | \9 / / 1 \ \9 /| \9 / 3 \/ 2 2
------- - 3*re|--------------------------------------| + I*|------- - 3*im|--------------------------------------| + -------------| - ------------- + ------- - 3*re|--------------------------------------| + I*|------- - 3*im|--------------------------------------| - -------------| + ------------- + - ---------------------- - ----------------------
2 | _________________| | 2 | _________________| 2 | 2 2 | _________________| | 2 | _________________| 2 | 2 _________________ 3
|/ ___\ / ___ | | |/ ___\ / ___ | | |/ ___\ / ___ | | |/ ___\ / ___ | | / ___
|| 1 I*\/ 3 | / 27 27*I*\/ 3 | | || 1 I*\/ 3 | / 27 27*I*\/ 3 | | || 1 I*\/ 3 | / 27 27*I*\/ 3 | | || 1 I*\/ 3 | / 27 27*I*\/ 3 | | / 27 27*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
$$\left(\left(\frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)}\right)\right) + \left(- \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} + i \left(- 3 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right)\right)\right) + \left(- \frac{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}{3} - \frac{3}{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)$$
=
_________________
/ ___
/ 27 27*I*\/ 3 / /pi\ ___ /pi\\ / /pi\ ___ /pi\\
3 / -- + ---------- |sin|--| \/ 3 *cos|--|| |sin|--| \/ 3 *cos|--||
3 / 1 \ / 1 \ \/ 2 2 | \9 / / 1 \ \9 /| | \9 / / 1 \ \9 /| /pi\
- ---------------------- - 3*re|--------------------------------------| - 3*re|--------------------------------------| - ---------------------- + I*|------- - 3*im|--------------------------------------| - -------------| + I*|------- - 3*im|--------------------------------------| + -------------| + cos|--|
_________________ | _________________| | _________________| 3 | 2 | _________________| 2 | | 2 | _________________| 2 | \9 /
/ ___ |/ ___\ / ___ | |/ ___\ / ___ | | |/ ___\ / ___ | | | |/ ___\ / ___ | |
/ 27 27*I*\/ 3 || 1 I*\/ 3 | / 27 27*I*\/ 3 | || 1 I*\/ 3 | / 27 27*I*\/ 3 | | || 1 I*\/ 3 | / 27 27*I*\/ 3 | | | || 1 I*\/ 3 | / 27 27*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 / /
$$- 3 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} - 3 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + \cos{\left(\frac{\pi}{9} \right)} - \frac{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}{3} + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)}\right) + i \left(- 3 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right) - \frac{3}{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}$$
producto
/ _________________\
| / ___ |
/ /pi\ / /pi\ ___ /pi\\ ___ /pi\\ / /pi\ / /pi\ ___ /pi\\ ___ /pi\\ | / 27 27*I*\/ 3 |
|cos|--| |sin|--| \/ 3 *cos|--|| \/ 3 *sin|--|| |cos|--| |sin|--| \/ 3 *cos|--|| \/ 3 *sin|--|| | 3 / -- + ---------- |
| \9 / / 1 \ | \9 / / 1 \ \9 /| \9 /| | \9 / / 1 \ | \9 / / 1 \ \9 /| \9 /| | 3 \/ 2 2 |
|------- - 3*re|--------------------------------------| + I*|------- - 3*im|--------------------------------------| + -------------| - -------------|*|------- - 3*re|--------------------------------------| + I*|------- - 3*im|--------------------------------------| - -------------| + -------------|*|- ---------------------- - ----------------------|
| 2 | _________________| | 2 | _________________| 2 | 2 | | 2 | _________________| | 2 | _________________| 2 | 2 | | _________________ 3 |
| |/ ___\ / ___ | | |/ ___\ / ___ | | | | |/ ___\ / ___ | | |/ ___\ / ___ | | | | / ___ |
| || 1 I*\/ 3 | / 27 27*I*\/ 3 | | || 1 I*\/ 3 | / 27 27*I*\/ 3 | | | | || 1 I*\/ 3 | / 27 27*I*\/ 3 | | || 1 I*\/ 3 | / 27 27*I*\/ 3 | | | | / 27 27*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 /
$$\left(- \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} + i \left(- 3 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right)\right) \left(\frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)}\right)\right) \left(- \frac{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}{3} - \frac{3}{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)$$
=
/ / 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 = 0.347296355333861
x3 = 1.53208888623796
x3 = 1.53208888623796
Gráfico