Sr Examen
3*x^3-12*x-8=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(3 x^{3} - 12 x\right) - 8 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - 4 x - \frac{8}{3} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -4$$
$$v = \frac{d}{a}$$
$$v = - \frac{8}{3}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -4$$
$$x_{1} x_{2} x_{3} = - \frac{8}{3}$$
$$\left(3 x^{3} - 12 x\right) - 8 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - 4 x - \frac{8}{3} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -4$$
$$v = \frac{d}{a}$$
$$v = - \frac{8}{3}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -4$$
$$x_{1} x_{2} x_{3} = - \frac{8}{3}$$
Suma y producto de raíces
[src]
suma
/ 1 \ / / 1 \ \ / 1 \ / / 1 \ \
2*re|----------------------------------| | 2*im|----------------------------------| | 2*re|----------------------------------| |2*im|----------------------------------| |
| _____________| | | _____________| | | _____________| | | _____________| |
|/ ___\ / ___ | | |/ ___\ / ___ | | |/ ___\ / ___ | | |/ ___\ / ___ | |
|| 1 I*\/ 3 | / 1 I*\/ 3 | | || 1 I*\/ 3 | / 1 I*\/ 3 | ___ /pi\| ___ /pi\ || 1 I*\/ 3 | / 1 I*\/ 3 | | || 1 I*\/ 3 | / 1 I*\/ 3 | ___ /pi\ | ___ /pi\ _____________
||- - - -------|*3 / - + ------- | | ||- - - -------|*3 / - + ------- | \/ 3 *sin|--|| \/ 3 *cos|--| ||- - + -------|*3 / - + ------- | | ||- - + -------|*3 / - + ------- | \/ 3 *sin|--| | \/ 3 *cos|--| / ___
\\ 2 2 / \/ 6 18 / | /pi\ \\ 2 2 / \/ 6 18 / \18/| \18/ /pi\ /pi\ \\ 2 2 / \/ 6 18 / | \\ 2 2 / \/ 6 18 / \18/ /pi\| \18/ / 1 I*\/ 3 2
---------------------------------------- + I*|- cos|--| + ---------------------------------------- - -------------| - ------------- + sin|--| + - sin|--| + ---------------------------------------- + I*|---------------------------------------- - ------------- + cos|--|| - ------------- + 2*3 / - + ------- + --------------------
3 \ \18/ 3 3 / 3 \18/ \18/ 3 \ 3 3 \18// 3 \/ 6 18 _____________
/ ___
/ 1 I*\/ 3
3*3 / - + -------
\/ 6 18
$$\left(\left(\frac{2 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3} - \frac{\sqrt{3} \cos{\left(\frac{\pi}{18} \right)}}{3} - \sin{\left(\frac{\pi}{18} \right)} + i \left(\frac{2 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{3} + \cos{\left(\frac{\pi}{18} \right)}\right)\right) + \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{18} \right)}}{3} + \frac{2 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3} + \sin{\left(\frac{\pi}{18} \right)} + i \left(- \cos{\left(\frac{\pi}{18} \right)} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{3} + \frac{2 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3}\right)\right)\right) + \left(\frac{2}{3 \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}} + 2 \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}\right)$$
=
/ 1 \ / 1 \ / / 1 \ \ / / 1 \ \
2*re|----------------------------------| 2*re|----------------------------------| | 2*im|----------------------------------| | |2*im|----------------------------------| |
| _____________| | _____________| | | _____________| | | | _____________| |
|/ ___\ / ___ | |/ ___\ / ___ | | |/ ___\ / ___ | | | |/ ___\ / ___ | |
_____________ || 1 I*\/ 3 | / 1 I*\/ 3 | || 1 I*\/ 3 | / 1 I*\/ 3 | | || 1 I*\/ 3 | / 1 I*\/ 3 | ___ /pi\| | || 1 I*\/ 3 | / 1 I*\/ 3 | ___ /pi\ | ___ /pi\
/ ___ ||- - + -------|*3 / - + ------- | ||- - - -------|*3 / - + ------- | | ||- - - -------|*3 / - + ------- | \/ 3 *sin|--|| | ||- - + -------|*3 / - + ------- | \/ 3 *sin|--| | 2*\/ 3 *cos|--|
/ 1 I*\/ 3 2 \\ 2 2 / \/ 6 18 / \\ 2 2 / \/ 6 18 / | /pi\ \\ 2 2 / \/ 6 18 / \18/| | \\ 2 2 / \/ 6 18 / \18/ /pi\| \18/
2*3 / - + ------- + -------------------- + ---------------------------------------- + ---------------------------------------- + I*|- cos|--| + ---------------------------------------- - -------------| + I*|---------------------------------------- - ------------- + cos|--|| - ---------------
\/ 6 18 _____________ 3 3 \ \18/ 3 3 / \ 3 3 \18// 3
/ ___
/ 1 I*\/ 3
3*3 / - + -------
\/ 6 18
$$- \frac{2 \sqrt{3} \cos{\left(\frac{\pi}{18} \right)}}{3} + \frac{2 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3} + \frac{2 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3} + \frac{2}{3 \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}} + i \left(\frac{2 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{3} + \cos{\left(\frac{\pi}{18} \right)}\right) + i \left(- \cos{\left(\frac{\pi}{18} \right)} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{3} + \frac{2 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3}\right) + 2 \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}$$
producto
/ / 1 \ / / 1 \ \ \ / / 1 \ / / 1 \ \ \
|2*re|----------------------------------| | 2*im|----------------------------------| | | | 2*re|----------------------------------| |2*im|----------------------------------| | |
| | _____________| | | _____________| | | | | _____________| | | _____________| | |
| |/ ___\ / ___ | | |/ ___\ / ___ | | | | |/ ___\ / ___ | | |/ ___\ / ___ | | |
| || 1 I*\/ 3 | / 1 I*\/ 3 | | || 1 I*\/ 3 | / 1 I*\/ 3 | ___ /pi\| ___ /pi\ | | || 1 I*\/ 3 | / 1 I*\/ 3 | | || 1 I*\/ 3 | / 1 I*\/ 3 | ___ /pi\ | ___ /pi\| / _____________ \
| ||- - - -------|*3 / - + ------- | | ||- - - -------|*3 / - + ------- | \/ 3 *sin|--|| \/ 3 *cos|--| | | ||- - + -------|*3 / - + ------- | | ||- - + -------|*3 / - + ------- | \/ 3 *sin|--| | \/ 3 *cos|--|| | / ___ |
| \\ 2 2 / \/ 6 18 / | /pi\ \\ 2 2 / \/ 6 18 / \18/| \18/ /pi\| | /pi\ \\ 2 2 / \/ 6 18 / | \\ 2 2 / \/ 6 18 / \18/ /pi\| \18/| | / 1 I*\/ 3 2 |
|---------------------------------------- + I*|- cos|--| + ---------------------------------------- - -------------| - ------------- + sin|--||*|- sin|--| + ---------------------------------------- + I*|---------------------------------------- - ------------- + cos|--|| - -------------|*|2*3 / - + ------- + --------------------|
\ 3 \ \18/ 3 3 / 3 \18// \ \18/ 3 \ 3 3 \18// 3 / | \/ 6 18 _____________|
| / ___ |
| / 1 I*\/ 3 |
| 3*3 / - + ------- |
\ \/ 6 18 /
$$\left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{18} \right)}}{3} + \frac{2 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3} + \sin{\left(\frac{\pi}{18} \right)} + i \left(- \cos{\left(\frac{\pi}{18} \right)} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{3} + \frac{2 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3}\right)\right) \left(\frac{2 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3} - \frac{\sqrt{3} \cos{\left(\frac{\pi}{18} \right)}}{3} - \sin{\left(\frac{\pi}{18} \right)} + i \left(\frac{2 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{3} + \cos{\left(\frac{\pi}{18} \right)}\right)\right) \left(\frac{2}{3 \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}} + 2 \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}\right)$$
=
/ 2/3\
/ /pi\\ | 3 ____ 3 ____ / ___\ |
|-12 + 24*cos|--||*\2*\/ 18 + \/ 12 *\3 + I*\/ 3 / /
\ \9 //
-------------------------------------------------------
_____________
3 / ___
27*\/ 3 + I*\/ 3
$$\frac{\left(-12 + 24 \cos{\left(\frac{\pi}{9} \right)}\right) \left(2 \sqrt[3]{18} + \sqrt[3]{12} \left(3 + \sqrt{3} i\right)^{\frac{2}{3}}\right)}{27 \sqrt[3]{3 + \sqrt{3} i}}$$
(-12 + 24*cos(pi/9))*(2*18^(1/3) + 12^(1/3)*(3 + i*sqrt(3))^(2/3))/(27*(3 + i*sqrt(3))^(1/3))
Respuesta rápida
[src]
/ 1 \ / / 1 \ \
2*re|----------------------------------| | 2*im|----------------------------------| |
| _____________| | | _____________| |
|/ ___\ / ___ | | |/ ___\ / ___ | |
|| 1 I*\/ 3 | / 1 I*\/ 3 | | || 1 I*\/ 3 | / 1 I*\/ 3 | ___ /pi\| ___ /pi\
||- - - -------|*3 / - + ------- | | ||- - - -------|*3 / - + ------- | \/ 3 *sin|--|| \/ 3 *cos|--|
\\ 2 2 / \/ 6 18 / | /pi\ \\ 2 2 / \/ 6 18 / \18/| \18/ /pi\
x1 = ---------------------------------------- + I*|- cos|--| + ---------------------------------------- - -------------| - ------------- + sin|--|
3 \ \18/ 3 3 / 3 \18/
$$x_{1} = - \frac{\sqrt{3} \cos{\left(\frac{\pi}{18} \right)}}{3} + \frac{2 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3} + \sin{\left(\frac{\pi}{18} \right)} + i \left(- \cos{\left(\frac{\pi}{18} \right)} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{3} + \frac{2 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3}\right)$$
/ 1 \ / / 1 \ \
2*re|----------------------------------| |2*im|----------------------------------| |
| _____________| | | _____________| |
|/ ___\ / ___ | | |/ ___\ / ___ | |
|| 1 I*\/ 3 | / 1 I*\/ 3 | | || 1 I*\/ 3 | / 1 I*\/ 3 | ___ /pi\ | ___ /pi\
||- - + -------|*3 / - + ------- | | ||- - + -------|*3 / - + ------- | \/ 3 *sin|--| | \/ 3 *cos|--|
/pi\ \\ 2 2 / \/ 6 18 / | \\ 2 2 / \/ 6 18 / \18/ /pi\| \18/
x2 = - sin|--| + ---------------------------------------- + I*|---------------------------------------- - ------------- + cos|--|| - -------------
\18/ 3 \ 3 3 \18// 3
$$x_{2} = \frac{2 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3} - \frac{\sqrt{3} \cos{\left(\frac{\pi}{18} \right)}}{3} - \sin{\left(\frac{\pi}{18} \right)} + i \left(\frac{2 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}}\right)}}{3} - \frac{\sqrt{3} \sin{\left(\frac{\pi}{18} \right)}}{3} + \cos{\left(\frac{\pi}{18} \right)}\right)$$
_____________
/ ___
/ 1 I*\/ 3 2
x3 = 2*3 / - + ------- + --------------------
\/ 6 18 _____________
/ ___
/ 1 I*\/ 3
3*3 / - + -------
\/ 6 18
$$x_{3} = \frac{2}{3 \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}} + 2 \sqrt[3]{\frac{1}{6} + \frac{\sqrt{3} i}{18}}$$
x3 = 2/(3*(1/6 + sqrt(3)*i/18)^(1/3)) + 2*(1/6 + sqrt(3)*i/18)^(1/3)
Respuesta numérica
[src]
x1 = -1.48445439793712
x2 = 2.27431608520652
x3 = -0.789861687269397
x3 = -0.789861687269397