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Expresiones semejantes
- (((x-1)^3)/3)+4*x=0
- (((x+1)^3)/3)-4*x=0
(((x-1)^3)/3)-4*x=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
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3 (x - 1) -------- - 4*x = 0 3
$$- 4 x + \frac{\left(x - 1\right)^{3}}{3} = 0$$
Respuesta rápida
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/ / ___\\ / / / ___\\ / / ___\\\ / / ___\\
| |\/ 7 || | | |\/ 7 || | |\/ 7 ||| | |\/ 7 ||
|atan|-----|| | |atan|-----|| |atan|-----||| |atan|-----||
| \ 3 /| / 1 \ | | \ 3 /| / 1 \ ___ | \ 3 /|| ___ | \ 3 /|
x1 = 1 - cos|-----------| + 4*re|----------------------------------| + I*|- sin|-----------| + 4*im|----------------------------------| - \/ 3 *cos|-----------|| + \/ 3 *sin|-----------|
\ 3 / | _______________ / ___\| | \ 3 / | _______________ / ___\| \ 3 /| \ 3 /
|3 / ___ | 1 I*\/ 3 || | |3 / ___ | 1 I*\/ 3 || |
|\/ 6 + 2*I*\/ 7 *|- - - -------|| | |\/ 6 + 2*I*\/ 7 *|- - - -------|| |
\ \ 2 2 // \ \ \ 2 2 // /
$$x_{1} = - \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} + 4 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{6 + 2 \sqrt{7} i}}\right)} + \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} + 1 + i \left(- \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} - \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} + 4 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{6 + 2 \sqrt{7} i}}\right)}\right)$$
/ / ___\\ / / / ___\\ / / ___\\\ / / ___\\
| |\/ 7 || | | |\/ 7 || | |\/ 7 ||| | |\/ 7 ||
|atan|-----|| | |atan|-----|| |atan|-----||| |atan|-----||
| \ 3 /| / 1 \ | | \ 3 /| / 1 \ ___ | \ 3 /|| ___ | \ 3 /|
x2 = 1 - cos|-----------| + 4*re|----------------------------------| + I*|- sin|-----------| + 4*im|----------------------------------| + \/ 3 *cos|-----------|| - \/ 3 *sin|-----------|
\ 3 / | _______________ / ___\| | \ 3 / | _______________ / ___\| \ 3 /| \ 3 /
|3 / ___ | 1 I*\/ 3 || | |3 / ___ | 1 I*\/ 3 || |
|\/ 6 + 2*I*\/ 7 *|- - + -------|| | |\/ 6 + 2*I*\/ 7 *|- - + -------|| |
\ \ 2 2 // \ \ \ 2 2 // /
$$x_{2} = 4 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{6 + 2 \sqrt{7} i}}\right)} - \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} - \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} + 1 + i \left(4 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{6 + 2 \sqrt{7} i}}\right)} - \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} + \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}\right)$$
_______________
3 / ___ 4
x3 = 1 + \/ 6 + 2*I*\/ 7 + ------------------
_______________
3 / ___
\/ 6 + 2*I*\/ 7
$$x_{3} = 1 + \frac{4}{\sqrt[3]{6 + 2 \sqrt{7} i}} + \sqrt[3]{6 + 2 \sqrt{7} i}$$
x3 = 1 + 4/(6 + 2*sqrt(7)*i)^(1/3) + (6 + 2*sqrt(7)*i)^(1/3)
Suma y producto de raíces
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suma
/ / ___\\ / / / ___\\ / / ___\\\ / / ___\\ / / ___\\ / / / ___\\ / / ___\\\ / / ___\\
| |\/ 7 || | | |\/ 7 || | |\/ 7 ||| | |\/ 7 || | |\/ 7 || | | |\/ 7 || | |\/ 7 ||| | |\/ 7 ||
|atan|-----|| | |atan|-----|| |atan|-----||| |atan|-----|| |atan|-----|| | |atan|-----|| |atan|-----||| |atan|-----|| _______________
| \ 3 /| / 1 \ | | \ 3 /| / 1 \ ___ | \ 3 /|| ___ | \ 3 /| | \ 3 /| / 1 \ | | \ 3 /| / 1 \ ___ | \ 3 /|| ___ | \ 3 /| 3 / ___ 4
1 - cos|-----------| + 4*re|----------------------------------| + I*|- sin|-----------| + 4*im|----------------------------------| - \/ 3 *cos|-----------|| + \/ 3 *sin|-----------| + 1 - cos|-----------| + 4*re|----------------------------------| + I*|- sin|-----------| + 4*im|----------------------------------| + \/ 3 *cos|-----------|| - \/ 3 *sin|-----------| + 1 + \/ 6 + 2*I*\/ 7 + ------------------
\ 3 / | _______________ / ___\| | \ 3 / | _______________ / ___\| \ 3 /| \ 3 / \ 3 / | _______________ / ___\| | \ 3 / | _______________ / ___\| \ 3 /| \ 3 / _______________
|3 / ___ | 1 I*\/ 3 || | |3 / ___ | 1 I*\/ 3 || | |3 / ___ | 1 I*\/ 3 || | |3 / ___ | 1 I*\/ 3 || | 3 / ___
|\/ 6 + 2*I*\/ 7 *|- - - -------|| | |\/ 6 + 2*I*\/ 7 *|- - - -------|| | |\/ 6 + 2*I*\/ 7 *|- - + -------|| | |\/ 6 + 2*I*\/ 7 *|- - + -------|| | \/ 6 + 2*I*\/ 7
\ \ 2 2 // \ \ \ 2 2 // / \ \ 2 2 // \ \ \ 2 2 // /
$$\left(1 + \frac{4}{\sqrt[3]{6 + 2 \sqrt{7} i}} + \sqrt[3]{6 + 2 \sqrt{7} i}\right) + \left(\left(- \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} + 4 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{6 + 2 \sqrt{7} i}}\right)} + \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} + 1 + i \left(- \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} - \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} + 4 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{6 + 2 \sqrt{7} i}}\right)}\right)\right) + \left(4 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{6 + 2 \sqrt{7} i}}\right)} - \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} - \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} + 1 + i \left(4 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{6 + 2 \sqrt{7} i}}\right)} - \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} + \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}\right)\right)\right)$$
=
/ / ___\\ / / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\
| |\/ 7 || | | |\/ 7 || | |\/ 7 ||| | | |\/ 7 || | |\/ 7 |||
_______________ |atan|-----|| | |atan|-----|| |atan|-----||| | |atan|-----|| |atan|-----|||
3 / ___ | \ 3 /| 4 / 1 \ / 1 \ | | \ 3 /| / 1 \ ___ | \ 3 /|| | | \ 3 /| / 1 \ ___ | \ 3 /||
3 + \/ 6 + 2*I*\/ 7 - 2*cos|-----------| + ------------------ + 4*re|----------------------------------| + 4*re|----------------------------------| + I*|- sin|-----------| + 4*im|----------------------------------| + \/ 3 *cos|-----------|| + I*|- sin|-----------| + 4*im|----------------------------------| - \/ 3 *cos|-----------||
\ 3 / _______________ | _______________ / ___\| | _______________ / ___\| | \ 3 / | _______________ / ___\| \ 3 /| | \ 3 / | _______________ / ___\| \ 3 /|
3 / ___ |3 / ___ | 1 I*\/ 3 || |3 / ___ | 1 I*\/ 3 || | |3 / ___ | 1 I*\/ 3 || | | |3 / ___ | 1 I*\/ 3 || |
\/ 6 + 2*I*\/ 7 |\/ 6 + 2*I*\/ 7 *|- - + -------|| |\/ 6 + 2*I*\/ 7 *|- - - -------|| | |\/ 6 + 2*I*\/ 7 *|- - + -------|| | | |\/ 6 + 2*I*\/ 7 *|- - - -------|| |
\ \ 2 2 // \ \ 2 2 // \ \ \ 2 2 // / \ \ \ 2 2 // /
$$- 2 \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} + 4 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{6 + 2 \sqrt{7} i}}\right)} + 4 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{6 + 2 \sqrt{7} i}}\right)} + 3 + \frac{4}{\sqrt[3]{6 + 2 \sqrt{7} i}} + i \left(- \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} - \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} + 4 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{6 + 2 \sqrt{7} i}}\right)}\right) + i \left(4 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{6 + 2 \sqrt{7} i}}\right)} - \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} + \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}\right) + \sqrt[3]{6 + 2 \sqrt{7} i}$$
producto
/ / / ___\\ / / / ___\\ / / ___\\\ / / ___\\\ / / / ___\\ / / / ___\\ / / ___\\\ / / ___\\\ | | |\/ 7 || | | |\/ 7 || | |\/ 7 ||| | |\/ 7 ||| | | |\/ 7 || | | |\/ 7 || | |\/ 7 ||| | |\/ 7 ||| | |atan|-----|| | |atan|-----|| |atan|-----||| |atan|-----||| | |atan|-----|| | |atan|-----|| |atan|-----||| |atan|-----||| / _______________ \ | | \ 3 /| / 1 \ | | \ 3 /| / 1 \ ___ | \ 3 /|| ___ | \ 3 /|| | | \ 3 /| / 1 \ | | \ 3 /| / 1 \ ___ | \ 3 /|| ___ | \ 3 /|| | 3 / ___ 4 | |1 - cos|-----------| + 4*re|----------------------------------| + I*|- sin|-----------| + 4*im|----------------------------------| - \/ 3 *cos|-----------|| + \/ 3 *sin|-----------||*|1 - cos|-----------| + 4*re|----------------------------------| + I*|- sin|-----------| + 4*im|----------------------------------| + \/ 3 *cos|-----------|| - \/ 3 *sin|-----------||*|1 + \/ 6 + 2*I*\/ 7 + ------------------| | \ 3 / | _______________ / ___\| | \ 3 / | _______________ / ___\| \ 3 /| \ 3 /| | \ 3 / | _______________ / ___\| | \ 3 / | _______________ / ___\| \ 3 /| \ 3 /| | _______________| | |3 / ___ | 1 I*\/ 3 || | |3 / ___ | 1 I*\/ 3 || | | | |3 / ___ | 1 I*\/ 3 || | |3 / ___ | 1 I*\/ 3 || | | | 3 / ___ | | |\/ 6 + 2*I*\/ 7 *|- - - -------|| | |\/ 6 + 2*I*\/ 7 *|- - - -------|| | | | |\/ 6 + 2*I*\/ 7 *|- - + -------|| | |\/ 6 + 2*I*\/ 7 *|- - + -------|| | | \ \/ 6 + 2*I*\/ 7 / \ \ \ 2 2 // \ \ \ 2 2 // / / \ \ \ 2 2 // \ \ \ 2 2 // / /
$$\left(- \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} + 4 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{6 + 2 \sqrt{7} i}}\right)} + \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} + 1 + i \left(- \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} - \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} + 4 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{6 + 2 \sqrt{7} i}}\right)}\right)\right) \left(4 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{6 + 2 \sqrt{7} i}}\right)} - \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} - \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} + 1 + i \left(4 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{6 + 2 \sqrt{7} i}}\right)} - \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} + \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}\right)\right) \left(1 + \frac{4}{\sqrt[3]{6 + 2 \sqrt{7} i}} + \sqrt[3]{6 + 2 \sqrt{7} i}\right)$$
=
/ / / ___\\ / / ___\\\
| | |\/ 7 || | |\/ 7 |||
/ _______________ / _______________\\ | |atan|-----|| |2*atan|-----|||
2/3 | 3 / ___ | 3 / ___ || | | \ 3 /| | \ 3 /||
2 *\4 + \/ 6 + 2*I*\/ 7 *\1 + \/ 6 + 2*I*\/ 7 //*|-3 - 4*cos|-----------| + 8*cos|-------------||
\ \ 3 / \ 3 //
-------------------------------------------------------------------------------------------------------
_____________
3 / ___
2*\/ 3 + I*\/ 7
$$\frac{2^{\frac{2}{3}} \left(4 + \left(1 + \sqrt[3]{6 + 2 \sqrt{7} i}\right) \sqrt[3]{6 + 2 \sqrt{7} i}\right) \left(- 4 \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} - 3 + 8 \cos{\left(\frac{2 \operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}\right)}{2 \sqrt[3]{3 + \sqrt{7} i}}$$
2^(2/3)*(4 + (6 + 2*i*sqrt(7))^(1/3)*(1 + (6 + 2*i*sqrt(7))^(1/3)))*(-3 - 4*cos(atan(sqrt(7)/3)/3) + 8*cos(2*atan(sqrt(7)/3)/3))/(2*(3 + i*sqrt(7))^(1/3))
Respuesta numérica
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x1 = -1.76873430527628
x2 = 4.88448370193933
x3 = -0.115749396663049
x3 = -0.115749396663049