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Expresiones semejantes
- x^3-1.2*x-0.5=0
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x^3-1.2*x+0.5=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
3 6*x 1
x - --- + - = 0
5 2
$$\left(x^{3} - \frac{6 x}{5}\right) + \frac{1}{2} = 0$$
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = - \frac{6}{5}$$
$$v = \frac{d}{a}$$
$$v = \frac{1}{2}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = - \frac{6}{5}$$
$$x_{1} x_{2} x_{3} = \frac{1}{2}$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = - \frac{6}{5}$$
$$v = \frac{d}{a}$$
$$v = \frac{1}{2}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = - \frac{6}{5}$$
$$x_{1} x_{2} x_{3} = \frac{1}{2}$$
Suma y producto de raíces
[src]
suma
/ 1 \ / / 1 \ \ / 1 \ / / 1 \ \
6*re|---------------------------------------| | 6*im|---------------------------------------| / / ____\\ / / ____\\| / / ____\\ / / ____\\ 6*re|---------------------------------------| | 6*im|---------------------------------------| / / ____\\ / / ____\\| / / ____\\ / / ____\\
| __________________| | | __________________| | |\/ 15 || | |\/ 15 ||| | |\/ 15 || | |\/ 15 || | __________________| | | __________________| | |\/ 15 || | |\/ 15 ||| | |\/ 15 || | |\/ 15 || __________________
|/ ___\ / ____ | | |/ ___\ / ____ | |atan|------|| |atan|------||| |atan|------|| |atan|------|| |/ ___\ / ____ | | |/ ___\ / ____ | |atan|------|| |atan|------||| |atan|------|| |atan|------|| / ____
|| 1 I*\/ 3 | / 27 27*I*\/ 15 | | || 1 I*\/ 3 | / 27 27*I*\/ 15 | ____ | \ 25 /| ____ | \ 25 /|| ____ | \ 25 /| ____ | \ 25 /| || 1 I*\/ 3 | / 27 27*I*\/ 15 | | || 1 I*\/ 3 | / 27 27*I*\/ 15 | ____ | \ 25 /| ____ | \ 25 /|| ____ | \ 25 /| ____ | \ 25 /| / 27 27*I*\/ 15
||- - - -------|*3 / -- + ----------- | | ||- - - -------|*3 / -- + ----------- | \/ 10 *sin|------------| \/ 30 *cos|------------|| \/ 30 *sin|------------| \/ 10 *cos|------------| ||- - + -------|*3 / -- + ----------- | | ||- - + -------|*3 / -- + ----------- | \/ 30 *cos|------------| \/ 10 *sin|------------|| \/ 10 *cos|------------| \/ 30 *sin|------------| 3 / -- + -----------
\\ 2 2 / \/ 4 100 / | \\ 2 2 / \/ 4 100 / \ 3 / \ 3 /| \ 3 / \ 3 / \\ 2 2 / \/ 4 100 / | \\ 2 2 / \/ 4 100 / \ 3 / \ 3 /| \ 3 / \ 3 / 6 \/ 4 100
- --------------------------------------------- + I*|- --------------------------------------------- + ------------------------ + ------------------------| - ------------------------ + ------------------------ + - --------------------------------------------- + I*|- --------------------------------------------- - ------------------------ + ------------------------| + ------------------------ + ------------------------ + - ------------------------- - -----------------------
5 \ 5 10 10 / 10 10 5 \ 5 10 10 / 10 10 __________________ 3
/ ____
/ 27 27*I*\/ 15
5*3 / -- + -----------
\/ 4 100
$$\left(\left(- \frac{\sqrt{30} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{10} - \frac{6 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{4} + \frac{27 \sqrt{15} i}{100}}}\right)}}{5} + \frac{\sqrt{10} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{10} + i \left(- \frac{6 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{4} + \frac{27 \sqrt{15} i}{100}}}\right)}}{5} + \frac{\sqrt{10} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{10} + \frac{\sqrt{30} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{10}\right)\right) + \left(\frac{\sqrt{30} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{10} + \frac{\sqrt{10} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{10} - \frac{6 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{4} + \frac{27 \sqrt{15} i}{100}}}\right)}}{5} + i \left(- \frac{\sqrt{30} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{10} + \frac{\sqrt{10} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{10} - \frac{6 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{4} + \frac{27 \sqrt{15} i}{100}}}\right)}}{5}\right)\right)\right) + \left(- \frac{\sqrt[3]{\frac{27}{4} + \frac{27 \sqrt{15} i}{100}}}{3} - \frac{6}{5 \sqrt[3]{\frac{27}{4} + \frac{27 \sqrt{15} i}{100}}}\right)$$
=
/ 1 \ / 1 \ / / 1 \ \ / / 1 \ \
6*re|---------------------------------------| 6*re|---------------------------------------| | 6*im|---------------------------------------| / / ____\\ / / ____\\| | 6*im|---------------------------------------| / / ____\\ / / ____\\| / / ____\\
| __________________| | __________________| __________________ | | __________________| | |\/ 15 || | |\/ 15 ||| | | __________________| | |\/ 15 || | |\/ 15 ||| | |\/ 15 ||
|/ ___\ / ____ | |/ ___\ / ____ | / ____ | |/ ___\ / ____ | |atan|------|| |atan|------||| | |/ ___\ / ____ | |atan|------|| |atan|------||| |atan|------||
|| 1 I*\/ 3 | / 27 27*I*\/ 15 | || 1 I*\/ 3 | / 27 27*I*\/ 15 | / 27 27*I*\/ 15 | || 1 I*\/ 3 | / 27 27*I*\/ 15 | ____ | \ 25 /| ____ | \ 25 /|| | || 1 I*\/ 3 | / 27 27*I*\/ 15 | ____ | \ 25 /| ____ | \ 25 /|| ____ | \ 25 /|
||- - + -------|*3 / -- + ----------- | ||- - - -------|*3 / -- + ----------- | 3 / -- + ----------- | ||- - + -------|*3 / -- + ----------- | \/ 30 *cos|------------| \/ 10 *sin|------------|| | ||- - - -------|*3 / -- + ----------- | \/ 10 *sin|------------| \/ 30 *cos|------------|| \/ 10 *cos|------------|
6 \\ 2 2 / \/ 4 100 / \\ 2 2 / \/ 4 100 / \/ 4 100 | \\ 2 2 / \/ 4 100 / \ 3 / \ 3 /| | \\ 2 2 / \/ 4 100 / \ 3 / \ 3 /| \ 3 /
- ------------------------- - --------------------------------------------- - --------------------------------------------- - ----------------------- + I*|- --------------------------------------------- - ------------------------ + ------------------------| + I*|- --------------------------------------------- + ------------------------ + ------------------------| + ------------------------
__________________ 5 5 3 \ 5 10 10 / \ 5 10 10 / 5
/ ____
/ 27 27*I*\/ 15
5*3 / -- + -----------
\/ 4 100
$$- \frac{6 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{4} + \frac{27 \sqrt{15} i}{100}}}\right)}}{5} - \frac{6 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{4} + \frac{27 \sqrt{15} i}{100}}}\right)}}{5} + \frac{\sqrt{10} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{5} - \frac{\sqrt[3]{\frac{27}{4} + \frac{27 \sqrt{15} i}{100}}}{3} + i \left(- \frac{\sqrt{30} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{10} + \frac{\sqrt{10} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{10} - \frac{6 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{4} + \frac{27 \sqrt{15} i}{100}}}\right)}}{5}\right) + i \left(- \frac{6 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{4} + \frac{27 \sqrt{15} i}{100}}}\right)}}{5} + \frac{\sqrt{10} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{10} + \frac{\sqrt{30} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{10}\right) - \frac{6}{5 \sqrt[3]{\frac{27}{4} + \frac{27 \sqrt{15} i}{100}}}$$
producto
/ / 1 \ / / 1 \ \ \ / / 1 \ / / 1 \ \ \
| 6*re|---------------------------------------| | 6*im|---------------------------------------| / / ____\\ / / ____\\| / / ____\\ / / ____\\| | 6*re|---------------------------------------| | 6*im|---------------------------------------| / / ____\\ / / ____\\| / / ____\\ / / ____\\|
| | __________________| | | __________________| | |\/ 15 || | |\/ 15 ||| | |\/ 15 || | |\/ 15 ||| | | __________________| | | __________________| | |\/ 15 || | |\/ 15 ||| | |\/ 15 || | |\/ 15 ||| / __________________\
| |/ ___\ / ____ | | |/ ___\ / ____ | |atan|------|| |atan|------||| |atan|------|| |atan|------||| | |/ ___\ / ____ | | |/ ___\ / ____ | |atan|------|| |atan|------||| |atan|------|| |atan|------||| | / ____ |
| || 1 I*\/ 3 | / 27 27*I*\/ 15 | | || 1 I*\/ 3 | / 27 27*I*\/ 15 | ____ | \ 25 /| ____ | \ 25 /|| ____ | \ 25 /| ____ | \ 25 /|| | || 1 I*\/ 3 | / 27 27*I*\/ 15 | | || 1 I*\/ 3 | / 27 27*I*\/ 15 | ____ | \ 25 /| ____ | \ 25 /|| ____ | \ 25 /| ____ | \ 25 /|| | / 27 27*I*\/ 15 |
| ||- - - -------|*3 / -- + ----------- | | ||- - - -------|*3 / -- + ----------- | \/ 10 *sin|------------| \/ 30 *cos|------------|| \/ 30 *sin|------------| \/ 10 *cos|------------|| | ||- - + -------|*3 / -- + ----------- | | ||- - + -------|*3 / -- + ----------- | \/ 30 *cos|------------| \/ 10 *sin|------------|| \/ 10 *cos|------------| \/ 30 *sin|------------|| | 3 / -- + ----------- |
| \\ 2 2 / \/ 4 100 / | \\ 2 2 / \/ 4 100 / \ 3 / \ 3 /| \ 3 / \ 3 /| | \\ 2 2 / \/ 4 100 / | \\ 2 2 / \/ 4 100 / \ 3 / \ 3 /| \ 3 / \ 3 /| | 6 \/ 4 100 |
|- --------------------------------------------- + I*|- --------------------------------------------- + ------------------------ + ------------------------| - ------------------------ + ------------------------|*|- --------------------------------------------- + I*|- --------------------------------------------- - ------------------------ + ------------------------| + ------------------------ + ------------------------|*|- ------------------------- - -----------------------|
\ 5 \ 5 10 10 / 10 10 / \ 5 \ 5 10 10 / 10 10 / | __________________ 3 |
| / ____ |
| / 27 27*I*\/ 15 |
| 5*3 / -- + ----------- |
\ \/ 4 100 /
$$\left(- \frac{\sqrt{30} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{10} - \frac{6 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{4} + \frac{27 \sqrt{15} i}{100}}}\right)}}{5} + \frac{\sqrt{10} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{10} + i \left(- \frac{6 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{4} + \frac{27 \sqrt{15} i}{100}}}\right)}}{5} + \frac{\sqrt{10} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{10} + \frac{\sqrt{30} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{10}\right)\right) \left(\frac{\sqrt{30} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{10} + \frac{\sqrt{10} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{10} - \frac{6 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{4} + \frac{27 \sqrt{15} i}{100}}}\right)}}{5} + i \left(- \frac{\sqrt{30} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{10} + \frac{\sqrt{10} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{10} - \frac{6 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{4} + \frac{27 \sqrt{15} i}{100}}}\right)}}{5}\right)\right) \left(- \frac{\sqrt[3]{\frac{27}{4} + \frac{27 \sqrt{15} i}{100}}}{3} - \frac{6}{5 \sqrt[3]{\frac{27}{4} + \frac{27 \sqrt{15} i}{100}}}\right)$$
=
/ / / ____\\\
| | |\/ 15 |||
| |2*atan|------||| / 2/3 \
3 ____ | | \ 25 /|| |/ ____\ 3 ____|
\/ 10 *|1 - 2*cos|--------------||*\\25 + I*\/ 15 / + 4*\/ 10 /
\ \ 3 //
------------------------------------------------------------------
_______________
3 / ____
25*\/ 25 + I*\/ 15
$$\frac{\sqrt[3]{10} \left(1 - 2 \cos{\left(\frac{2 \operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}\right) \left(4 \sqrt[3]{10} + \left(25 + \sqrt{15} i\right)^{\frac{2}{3}}\right)}{25 \sqrt[3]{25 + \sqrt{15} i}}$$
10^(1/3)*(1 - 2*cos(2*atan(sqrt(15)/25)/3))*((25 + i*sqrt(15))^(2/3) + 4*10^(1/3))/(25*(25 + i*sqrt(15))^(1/3))
Respuesta rápida
[src]
/ 1 \ / / 1 \ \
6*re|---------------------------------------| | 6*im|---------------------------------------| / / ____\\ / / ____\\| / / ____\\ / / ____\\
| __________________| | | __________________| | |\/ 15 || | |\/ 15 ||| | |\/ 15 || | |\/ 15 ||
|/ ___\ / ____ | | |/ ___\ / ____ | |atan|------|| |atan|------||| |atan|------|| |atan|------||
|| 1 I*\/ 3 | / 27 27*I*\/ 15 | | || 1 I*\/ 3 | / 27 27*I*\/ 15 | ____ | \ 25 /| ____ | \ 25 /|| ____ | \ 25 /| ____ | \ 25 /|
||- - - -------|*3 / -- + ----------- | | ||- - - -------|*3 / -- + ----------- | \/ 10 *sin|------------| \/ 30 *cos|------------|| \/ 30 *sin|------------| \/ 10 *cos|------------|
\\ 2 2 / \/ 4 100 / | \\ 2 2 / \/ 4 100 / \ 3 / \ 3 /| \ 3 / \ 3 /
x1 = - --------------------------------------------- + I*|- --------------------------------------------- + ------------------------ + ------------------------| - ------------------------ + ------------------------
5 \ 5 10 10 / 10 10
$$x_{1} = - \frac{\sqrt{30} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{10} - \frac{6 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{4} + \frac{27 \sqrt{15} i}{100}}}\right)}}{5} + \frac{\sqrt{10} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{10} + i \left(- \frac{6 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{4} + \frac{27 \sqrt{15} i}{100}}}\right)}}{5} + \frac{\sqrt{10} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{10} + \frac{\sqrt{30} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{10}\right)$$
/ 1 \ / / 1 \ \
6*re|---------------------------------------| | 6*im|---------------------------------------| / / ____\\ / / ____\\| / / ____\\ / / ____\\
| __________________| | | __________________| | |\/ 15 || | |\/ 15 ||| | |\/ 15 || | |\/ 15 ||
|/ ___\ / ____ | | |/ ___\ / ____ | |atan|------|| |atan|------||| |atan|------|| |atan|------||
|| 1 I*\/ 3 | / 27 27*I*\/ 15 | | || 1 I*\/ 3 | / 27 27*I*\/ 15 | ____ | \ 25 /| ____ | \ 25 /|| ____ | \ 25 /| ____ | \ 25 /|
||- - + -------|*3 / -- + ----------- | | ||- - + -------|*3 / -- + ----------- | \/ 30 *cos|------------| \/ 10 *sin|------------|| \/ 10 *cos|------------| \/ 30 *sin|------------|
\\ 2 2 / \/ 4 100 / | \\ 2 2 / \/ 4 100 / \ 3 / \ 3 /| \ 3 / \ 3 /
x2 = - --------------------------------------------- + I*|- --------------------------------------------- - ------------------------ + ------------------------| + ------------------------ + ------------------------
5 \ 5 10 10 / 10 10
$$x_{2} = \frac{\sqrt{30} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{10} + \frac{\sqrt{10} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{10} - \frac{6 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{4} + \frac{27 \sqrt{15} i}{100}}}\right)}}{5} + i \left(- \frac{\sqrt{30} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{10} + \frac{\sqrt{10} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{15}}{25} \right)}}{3} \right)}}{10} - \frac{6 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{4} + \frac{27 \sqrt{15} i}{100}}}\right)}}{5}\right)$$
__________________
/ ____
/ 27 27*I*\/ 15
3 / -- + -----------
6 \/ 4 100
x3 = - ------------------------- - -----------------------
__________________ 3
/ ____
/ 27 27*I*\/ 15
5*3 / -- + -----------
\/ 4 100
$$x_{3} = - \frac{\sqrt[3]{\frac{27}{4} + \frac{27 \sqrt{15} i}{100}}}{3} - \frac{6}{5 \sqrt[3]{\frac{27}{4} + \frac{27 \sqrt{15} i}{100}}}$$
x3 = -(27/4 + 27*sqrt(15)*i/100)^(1/3)/3 - 6/(5*(27/4 + 27*sqrt(15)*i/100)^(1/3))
Respuesta numérica
[src]
x1 = 0.575527835434157
x2 = -1.26325137687806
x3 = 0.687723541443907
x3 = 0.687723541443907