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- La ecuación:
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Ecuación -x-2+3*(x-3)=3*(4-x)-3
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Ecuación 2x=18-x
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Ecuación tan(x)=-1
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Ecuación 17x-8=20x+7
- Expresar {x} en función de y en la ecuación:
- 9*x-4*y=2
- 14*x+16*y=14
- -1*x+17*y=-12
- 15*x+15*y=16
- Gráfico de la función y =:
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x^3-3*x^2+1
-
Expresiones idénticas
- x^ tres - tres *x^ dos + uno = cero
- x al cubo menos 3 multiplicar por x al cuadrado más 1 es igual a 0
- x en el grado tres menos tres multiplicar por x en el grado dos más uno es igual a cero
- x3-3*x2+1=0
- x³-3*x²+1=0
- x en el grado 3-3*x en el grado 2+1=0
- x^3-3x^2+1=0
- x3-3x2+1=0
- x^3-3*x^2+1=O
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Expresiones semejantes
- x^3+3*x^2+1=0
- x^3-3*x^2-1=0
x^3-3*x^2+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
$$p = \frac{b}{a}$$
$$p = -3$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = 1$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 3$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = 1$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -3$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = 1$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 3$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = 1$$
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
1 - ------- + I*|- ------- - ------------- + im|----------------------------------|| + ------------- + re|----------------------------------| + 1 - ------- + I*|- ------- + ------------- + im|----------------------------------|| - ------------- + re|----------------------------------| + 1 + ------------------ + 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
$$\left(\left(\operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} - \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + 1 + i \left(\operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right)\right) + \left(- \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + 1 + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)}\right)\right)\right) + \left(1 + \frac{1}{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}} + \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}\right)$$
=
_____________ / /pi\ ___ /pi\ \ / /pi\ ___ /pi\ \
/ ___ | sin|--| \/ 3 *cos|--| | | sin|--| \/ 3 *cos|--| |
1 / 1 I*\/ 3 /pi\ | \9 / \9 / / 1 \| | \9 / \9 / / 1 \| / 1 \ / 1 \
3 + ------------------ + 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 //
$$- \cos{\left(\frac{\pi}{9} \right)} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} + 3 + \frac{1}{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}} + i \left(\operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right) + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)}\right) + \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}$$
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 | |1 - ------- + I*|- ------- - ------------- + im|----------------------------------|| + ------------- + re|----------------------------------||*|1 - ------- + I*|- ------- + ------------- + im|----------------------------------|| - ------------- + re|----------------------------------||*|1 + ------------------ + 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 /
$$\left(- \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + 1 + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)}\right)\right) \left(\operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} - \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + 1 + i \left(\operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right)\right) \left(1 + \frac{1}{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}} + \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}\right)$$
=
/ _____________ / _____________\\
/ /pi\\ | 3 ___ 3 / ___ | 2/3 3 / ___ || / ___ /pi\ /pi\\
|-2 + 4*sin|--||*\2*\/ 2 + \/ 1 + I*\/ 3 *\2 + 2 *\/ 1 + I*\/ 3 //*|-1 + \/ 3 *sin|--| + cos|--||
\ \18// \ \9 / \9 //
--------------------------------------------------------------------------------------------------------
_____________
3 / ___
4*\/ 1 + I*\/ 3
$$\frac{\left(-2 + 4 \sin{\left(\frac{\pi}{18} \right)}\right) \left(2 \sqrt[3]{2} + \sqrt[3]{1 + \sqrt{3} i} \left(2 + 2^{\frac{2}{3}} \sqrt[3]{1 + \sqrt{3} i}\right)\right) \left(-1 + \sqrt{3} \sin{\left(\frac{\pi}{9} \right)} + \cos{\left(\frac{\pi}{9} \right)}\right)}{4 \sqrt[3]{1 + \sqrt{3} i}}$$
(-2 + 4*sin(pi/18))*(2*2^(1/3) + (1 + i*sqrt(3))^(1/3)*(2 + 2^(2/3)*(1 + i*sqrt(3))^(1/3)))*(-1 + sqrt(3)*sin(pi/9) + cos(pi/9))/(4*(1 + i*sqrt(3))^(1/3))
Respuesta rápida
[src]
/pi\ / /pi\ ___ /pi\ \ ___ /pi\
cos|--| | sin|--| \/ 3 *cos|--| | \/ 3 *sin|--|
\9 / | \9 / \9 / / 1 \| \9 / / 1 \
x1 = 1 - ------- + 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 //
$$x_{1} = - \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + 1 + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)}\right)$$
/pi\ / /pi\ ___ /pi\ \ ___ /pi\
cos|--| | sin|--| \/ 3 *cos|--| | \/ 3 *sin|--|
\9 / | \9 / \9 / / 1 \| \9 / / 1 \
x2 = 1 - ------- + 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 //
$$x_{2} = \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} - \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + 1 + i \left(\operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}}\right)} - \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right)$$
_____________
/ ___
1 / 1 I*\/ 3
x3 = 1 + ------------------ + 3 / - + -------
_____________ \/ 2 2
/ ___
/ 1 I*\/ 3
3 / - + -------
\/ 2 2
$$x_{3} = 1 + \frac{1}{\sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}} + \sqrt[3]{\frac{1}{2} + \frac{\sqrt{3} i}{2}}$$
x3 = 1 + (1/2 + sqrt(3)*i/2)^(-1/3) + (1/2 + sqrt(3)*i/2)^(1/3)
Respuesta numérica
[src]
x1 = -0.532088886237956
x2 = 2.87938524157182
x3 = 0.652703644666139
x3 = 0.652703644666139
Gráfico