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x^3-81(x+3)=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
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -81$$
$$v = \frac{d}{a}$$
$$v = -243$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -81$$
$$x_{1} x_{2} x_{3} = -243$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -81$$
$$v = \frac{d}{a}$$
$$v = -243$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -81$$
$$x_{1} x_{2} x_{3} = -243$$
Respuesta rápida
[src]
/pi\ / /pi\ ___ /pi\\ ___ /pi\
9*sin|--| | 9*cos|--| 3*\/ 3 *sin|--|| 3*\/ 3 *cos|--|
/ 1 \ \18/ | / 1 \ \18/ \18/| \18/
x1 = 9*re|------------------------------------| + --------- + I*|9*im|------------------------------------| - --------- - ---------------| - ---------------
| _______________| 2 | | _______________| 2 2 | 2
|/ ___\ / ___ | | |/ ___\ / ___ | |
|| 1 I*\/ 3 | / 9 3*I*\/ 3 | | || 1 I*\/ 3 | / 9 3*I*\/ 3 | |
||- - - -------|*3 / - + --------- | | ||- - - -------|*3 / - + --------- | |
\\ 2 2 / \/ 2 2 / \ \\ 2 2 / \/ 2 2 / /
$$x_{1} = - \frac{3 \sqrt{3} \cos{\left(\frac{\pi}{18} \right)}}{2} + 9 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)} + \frac{9 \sin{\left(\frac{\pi}{18} \right)}}{2} + i \left(- \frac{9 \cos{\left(\frac{\pi}{18} \right)}}{2} - \frac{3 \sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{2} + 9 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)}\right)$$
/pi\ / /pi\ ___ /pi\\ ___ /pi\
9*sin|--| | 9*cos|--| 3*\/ 3 *sin|--|| 3*\/ 3 *cos|--|
/ 1 \ \18/ | / 1 \ \18/ \18/| \18/
x2 = 9*re|------------------------------------| - --------- + I*|9*im|------------------------------------| + --------- - ---------------| - ---------------
| _______________| 2 | | _______________| 2 2 | 2
|/ ___\ / ___ | | |/ ___\ / ___ | |
|| 1 I*\/ 3 | / 9 3*I*\/ 3 | | || 1 I*\/ 3 | / 9 3*I*\/ 3 | |
||- - + -------|*3 / - + --------- | | ||- - + -------|*3 / - + --------- | |
\\ 2 2 / \/ 2 2 / \ \\ 2 2 / \/ 2 2 / /
$$x_{2} = 9 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)} - \frac{3 \sqrt{3} \cos{\left(\frac{\pi}{18} \right)}}{2} - \frac{9 \sin{\left(\frac{\pi}{18} \right)}}{2} + i \left(9 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)} - \frac{3 \sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{2} + \frac{9 \cos{\left(\frac{\pi}{18} \right)}}{2}\right)$$
_______________
/ ___
/ 9 3*I*\/ 3 9
x3 = 3*3 / - + --------- + --------------------
\/ 2 2 _______________
/ ___
/ 9 3*I*\/ 3
3 / - + ---------
\/ 2 2
$$x_{3} = \frac{9}{\sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}} + 3 \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}$$
x3 = 9/(9/2 + 3*sqrt(3)*i/2)^(1/3) + 3*(9/2 + 3*sqrt(3)*i/2)^(1/3)
Suma y producto de raíces
[src]
suma
/pi\ / /pi\ ___ /pi\\ ___ /pi\ /pi\ / /pi\ ___ /pi\\ ___ /pi\ _______________
9*sin|--| | 9*cos|--| 3*\/ 3 *sin|--|| 3*\/ 3 *cos|--| 9*sin|--| | 9*cos|--| 3*\/ 3 *sin|--|| 3*\/ 3 *cos|--| / ___
/ 1 \ \18/ | / 1 \ \18/ \18/| \18/ / 1 \ \18/ | / 1 \ \18/ \18/| \18/ / 9 3*I*\/ 3 9
9*re|------------------------------------| + --------- + I*|9*im|------------------------------------| - --------- - ---------------| - --------------- + 9*re|------------------------------------| - --------- + I*|9*im|------------------------------------| + --------- - ---------------| - --------------- + 3*3 / - + --------- + --------------------
| _______________| 2 | | _______________| 2 2 | 2 | _______________| 2 | | _______________| 2 2 | 2 \/ 2 2 _______________
|/ ___\ / ___ | | |/ ___\ / ___ | | |/ ___\ / ___ | | |/ ___\ / ___ | | / ___
|| 1 I*\/ 3 | / 9 3*I*\/ 3 | | || 1 I*\/ 3 | / 9 3*I*\/ 3 | | || 1 I*\/ 3 | / 9 3*I*\/ 3 | | || 1 I*\/ 3 | / 9 3*I*\/ 3 | | / 9 3*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
$$\left(\left(9 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)} - \frac{3 \sqrt{3} \cos{\left(\frac{\pi}{18} \right)}}{2} - \frac{9 \sin{\left(\frac{\pi}{18} \right)}}{2} + i \left(9 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)} - \frac{3 \sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{2} + \frac{9 \cos{\left(\frac{\pi}{18} \right)}}{2}\right)\right) + \left(- \frac{3 \sqrt{3} \cos{\left(\frac{\pi}{18} \right)}}{2} + 9 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)} + \frac{9 \sin{\left(\frac{\pi}{18} \right)}}{2} + i \left(- \frac{9 \cos{\left(\frac{\pi}{18} \right)}}{2} - \frac{3 \sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{2} + 9 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)}\right)\right)\right) + \left(\frac{9}{\sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}} + 3 \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}\right)$$
=
_______________ / /pi\ ___ /pi\\ / /pi\ ___ /pi\\
/ ___ | 9*cos|--| 3*\/ 3 *sin|--|| | 9*cos|--| 3*\/ 3 *sin|--||
/ 9 3*I*\/ 3 9 / 1 \ / 1 \ | / 1 \ \18/ \18/| | / 1 \ \18/ \18/| ___ /pi\
3*3 / - + --------- + -------------------- + 9*re|------------------------------------| + 9*re|------------------------------------| + I*|9*im|------------------------------------| + --------- - ---------------| + I*|9*im|------------------------------------| - --------- - ---------------| - 3*\/ 3 *cos|--|
\/ 2 2 _______________ | _______________| | _______________| | | _______________| 2 2 | | | _______________| 2 2 | \18/
/ ___ |/ ___\ / ___ | |/ ___\ / ___ | | |/ ___\ / ___ | | | |/ ___\ / ___ | |
/ 9 3*I*\/ 3 || 1 I*\/ 3 | / 9 3*I*\/ 3 | || 1 I*\/ 3 | / 9 3*I*\/ 3 | | || 1 I*\/ 3 | / 9 3*I*\/ 3 | | | || 1 I*\/ 3 | / 9 3*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 / /
$$- 3 \sqrt{3} \cos{\left(\frac{\pi}{18} \right)} + 9 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)} + 9 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)} + \frac{9}{\sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}} + i \left(- \frac{9 \cos{\left(\frac{\pi}{18} \right)}}{2} - \frac{3 \sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{2} + 9 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)}\right) + i \left(9 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)} - \frac{3 \sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{2} + \frac{9 \cos{\left(\frac{\pi}{18} \right)}}{2}\right) + 3 \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}$$
producto
/ /pi\ / /pi\ ___ /pi\\ ___ /pi\\ / /pi\ / /pi\ ___ /pi\\ ___ /pi\\ / _______________ \ | 9*sin|--| | 9*cos|--| 3*\/ 3 *sin|--|| 3*\/ 3 *cos|--|| | 9*sin|--| | 9*cos|--| 3*\/ 3 *sin|--|| 3*\/ 3 *cos|--|| | / ___ | | / 1 \ \18/ | / 1 \ \18/ \18/| \18/| | / 1 \ \18/ | / 1 \ \18/ \18/| \18/| | / 9 3*I*\/ 3 9 | |9*re|------------------------------------| + --------- + I*|9*im|------------------------------------| - --------- - ---------------| - ---------------|*|9*re|------------------------------------| - --------- + I*|9*im|------------------------------------| + --------- - ---------------| - ---------------|*|3*3 / - + --------- + --------------------| | | _______________| 2 | | _______________| 2 2 | 2 | | | _______________| 2 | | _______________| 2 2 | 2 | | \/ 2 2 _______________| | |/ ___\ / ___ | | |/ ___\ / ___ | | | | |/ ___\ / ___ | | |/ ___\ / ___ | | | | / ___ | | || 1 I*\/ 3 | / 9 3*I*\/ 3 | | || 1 I*\/ 3 | / 9 3*I*\/ 3 | | | | || 1 I*\/ 3 | / 9 3*I*\/ 3 | | || 1 I*\/ 3 | / 9 3*I*\/ 3 | | | | / 9 3*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 /
$$\left(- \frac{3 \sqrt{3} \cos{\left(\frac{\pi}{18} \right)}}{2} + 9 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)} + \frac{9 \sin{\left(\frac{\pi}{18} \right)}}{2} + i \left(- \frac{9 \cos{\left(\frac{\pi}{18} \right)}}{2} - \frac{3 \sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{2} + 9 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)}\right)\right) \left(9 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)} - \frac{3 \sqrt{3} \cos{\left(\frac{\pi}{18} \right)}}{2} - \frac{9 \sin{\left(\frac{\pi}{18} \right)}}{2} + i \left(9 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}}\right)} - \frac{3 \sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{2} + \frac{9 \cos{\left(\frac{\pi}{18} \right)}}{2}\right)\right) \left(\frac{9}{\sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}} + 3 \sqrt[3]{\frac{9}{2} + \frac{3 \sqrt{3} i}{2}}\right)$$
=
/ /pi\\
| cos|--|| / 2/3 \
2/3 | 1 \9 /| |/ ___\ 3 ___|
54*3 *|- - + -------|*\\18 + 6*I*\/ 3 / + 6*\/ 2 /
\ 4 2 /
-------------------------------------------------------
_____________
3 / ___
\/ 3 + I*\/ 3
$$\frac{54 \cdot 3^{\frac{2}{3}} \left(- \frac{1}{4} + \frac{\cos{\left(\frac{\pi}{9} \right)}}{2}\right) \left(6 \sqrt[3]{2} + \left(18 + 6 \sqrt{3} i\right)^{\frac{2}{3}}\right)}{\sqrt[3]{3 + \sqrt{3} i}}$$
54*3^(2/3)*(-1/4 + cos(pi/9)/2)*((18 + 6*i*sqrt(3))^(2/3) + 6*2^(1/3))/(3 + i*sqrt(3))^(1/3)
Respuesta numérica
[src]
x1 = 10.2344223834293
x2 = -3.55437759271229
x3 = -6.68004479071703
x3 = -6.68004479071703