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- La ecuación:
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Ecuación 10=-2-4x
- Ecuación (243*y^2)^(1/3)=18+(81y)^(1/3)
- Ecuación -1=5x+1
- Ecuación 0,5^7-3x=4
- Expresar {x} en función de y en la ecuación:
- -8*x-6*y=-20
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- -6*x-8*y=18
- -11*x-18*y=18
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Expresiones idénticas
- x^ cuatro + tres x^3- cuatro y^ dos -18y+4= cero
- x en el grado 4 más 3x al cubo menos 4y al cuadrado menos 18y más 4 es igual a 0
- x en el grado cuatro más tres x al cubo menos cuatro y en el grado dos menos 18y más 4 es igual a cero
- x4+3x3-4y2-18y+4=0
- x⁴+3x³-4y²-18y+4=0
- x en el grado 4+3x en el grado 3-4y en el grado 2-18y+4=0
- x^4+3x^3-4y^2-18y+4=O
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Expresiones semejantes
- x^4+3x^3+4y^2-18y+4=0
- x^4-3x^3-4y^2-18y+4=0
- x^4+3x^3-4y^2-18y-4=0
- x^4+3x^3-4y^2+18y+4=0
x^4+3x^3-4y^2-18y+4=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
4 3 2 x + 3*x - 4*y - 18*y + 4 = 0
$$\left(- 18 y + \left(- 4 y^{2} + \left(x^{4} + 3 x^{3}\right)\right)\right) + 4 = 0$$
Gráfica
Respuesta rápida
[src]
// ___________________________________________________________________________ \ // ___________________________________________________________________________ \
|| / ___________________ | || / ___________________ |
|| / / 2 | || / / 2 |
|| / 9 / 9 9*y 81*y 27 | || / 9 / 9 9*y 81*y 27 |
|| / - + 2*3 / - - + ---- + ---- + ---------------------------------------- | || / - + 2*3 / - - + ---- + ---- + ---------------------------------------- |
|| ________________________________ / 2 \/ 2 2 4 ________________________________ | || ________________________________ / 2 \/ 2 2 4 ________________________________ |
|| / ___________________ / / ___________________ | || / ___________________ / / ___________________ |
|| / / 2 / / / 2 | || / / 2 / / / 2 |
|| / 9 / 9 9*y 81*y / / 9 / 9 9*y 81*y | || / 9 / 9 9*y 81*y / / 9 / 9 9*y 81*y |
|| / - - 2*3 / - - + ---- + ---- / 4* / - - 2*3 / - - + ---- + ---- | || / - - 2*3 / - - + ---- + ---- / 4* / - - 2*3 / - - + ---- + ---- |
|| 3 \/ 4 \/ 2 2 4 \/ \/ 4 \/ 2 2 4 2 | || 3 \/ 4 \/ 2 2 4 \/ \/ 4 \/ 2 2 4 2 |
|| - - - -------------------------------------- - -------------------------------------------------------------------------------------- for -4 + 4*y + 18*y = 0| || - - - -------------------------------------- - -------------------------------------------------------------------------------------- for -4 + 4*y + 18*y = 0|
|| 4 2 2 | || 4 2 2 |
|| | || |
|| _____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ | || _____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ |
|| / ________________________________________________________________________ | || / ________________________________________________________________________ |
|| / / ____________________________________________ | || / / ____________________________________________ |
|| / / / 2 | || / / / 2 |
|| / / / / 2 \ | || / / / / 2 \ |
|| / / / | 9 9*y 81*y| 3 | || / / / | 9 9*y 81*y| 3 |
|| / / / |- - + ---- + ----| / 2 \ 2 / 2 \ | || / / / |- - + ---- + ----| / 2 \ 2 / 2 \ |
x1 = I*im|< / 9 / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y 27 2*\-4 + 4*y + 18*y/ | + re|< / 9 / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y 27 2*\-4 + 4*y + 18*y/ |
|| / - - 2*3 / - + / -------------------- + ------------------- - ---- - ---- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------- | || / - - 2*3 / - + / -------------------- + ------------------- - ---- - ---- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------- |
|| _______________________________________________________________________________________________________________________________________________________________________________ / 2 \/ 4 \/ 4 27 8 4 _______________________________________________________________________________________________________________________________________________________________________________ ________________________________________________________________________ | || _______________________________________________________________________________________________________________________________________________________________________________ / 2 \/ 4 \/ 4 27 8 4 _______________________________________________________________________________________________________________________________________________________________________________ ________________________________________________________________________ |
|| / ________________________________________________________________________ / / ________________________________________________________________________ / ____________________________________________ | || / ________________________________________________________________________ / / ________________________________________________________________________ / ____________________________________________ |
|| / / ____________________________________________ / / / ____________________________________________ / / 2 | || / / ____________________________________________ / / / ____________________________________________ / / 2 |
|| / / / 2 / / / / 2 / / / 2 \ | || / / / 2 / / / / 2 / / / 2 \ |
|| / / / / 2 \ / / / / / 2 \ / / | 9 9*y 81*y| 3 | || / / / / 2 \ / / / / / 2 \ / / | 9 9*y 81*y| 3 |
|| / / / | 9 9*y 81*y| 3 / / / / | 9 9*y 81*y| 3 / / |- - + ---- + ----| / 2 \ 2 | || / / / | 9 9*y 81*y| 3 / / / / | 9 9*y 81*y| 3 / / |- - + ---- + ----| / 2 \ 2 |
|| / / / |- - + ---- + ----| / 2 \ 2 / 2 \ / / / / |- - + ---- + ----| / 2 \ 2 / 2 \ / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y | || / / / |- - + ---- + ----| / 2 \ 2 / 2 \ / / / / |- - + ---- + ----| / 2 \ 2 / 2 \ / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y |
|| / 9 / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y 2*\-4 + 4*y + 18*y/ / / 9 / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y 2*\-4 + 4*y + 18*y/ 3*3 / - + / -------------------- + ------------------- - ---- - ---- | || / 9 / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y 2*\-4 + 4*y + 18*y/ / / 9 / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y 2*\-4 + 4*y + 18*y/ 3*3 / - + / -------------------- + ------------------- - ---- - ---- |
|| / - + 2*3 / - + / -------------------- + ------------------- - ---- - ---- - ----------------------------------------------------------------------------------- / 4* / - + 2*3 / - + / -------------------- + ------------------- - ---- - ---- - ----------------------------------------------------------------------------------- \/ 4 \/ 4 27 8 4 | || / - + 2*3 / - + / -------------------- + ------------------- - ---- - ---- - ----------------------------------------------------------------------------------- / 4* / - + 2*3 / - + / -------------------- + ------------------- - ---- - ---- - ----------------------------------------------------------------------------------- \/ 4 \/ 4 27 8 4 |
|| / 4 \/ 4 \/ 4 27 8 4 ________________________________________________________________________ / / 4 \/ 4 \/ 4 27 8 4 ________________________________________________________________________ | || / 4 \/ 4 \/ 4 27 8 4 ________________________________________________________________________ / / 4 \/ 4 \/ 4 27 8 4 ________________________________________________________________________ |
|| / / ____________________________________________ / / / ____________________________________________ | || / / ____________________________________________ / / / ____________________________________________ |
|| / / / 2 / / / / 2 | || / / / 2 / / / / 2 |
|| / / / / 2 \ / / / / / 2 \ | || / / / / 2 \ / / / / / 2 \ |
|| / / / | 9 9*y 81*y| 3 / / / / | 9 9*y 81*y| 3 | || / / / | 9 9*y 81*y| 3 / / / / | 9 9*y 81*y| 3 |
|| / / / |- - + ---- + ----| / 2 \ 2 / / / / |- - + ---- + ----| / 2 \ 2 | || / / / |- - + ---- + ----| / 2 \ 2 / / / / |- - + ---- + ----| / 2 \ 2 |
|| / / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y / / / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y | || / / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y / / / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y |
|| / 3*3 / - + / -------------------- + ------------------- - ---- - ---- / / 3*3 / - + / -------------------- + ------------------- - ---- - ---- | || / 3*3 / - + / -------------------- + ------------------- - ---- - ---- / / 3*3 / - + / -------------------- + ------------------- - ---- - ---- |
|| 3 \/ \/ 4 \/ 4 27 8 4 \/ \/ \/ 4 \/ 4 27 8 4 | || 3 \/ \/ 4 \/ 4 27 8 4 \/ \/ \/ 4 \/ 4 27 8 4 |
||- - - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- otherwise | ||- - - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- otherwise |
\\ 4 2 2 / \\ 4 2 2 /
$$x_{1} = \operatorname{re}{\left(\begin{cases} - \frac{\sqrt{\frac{9}{4} - 2 \sqrt[3]{\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}}}}{2} - \frac{\sqrt{2 \sqrt[3]{\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}} + \frac{9}{2} + \frac{27}{4 \sqrt{\frac{9}{4} - 2 \sqrt[3]{\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}}}}}}{2} - \frac{3}{4} & \text{for}\: 4 y^{2} + 18 y - 4 = 0 \\- \frac{\sqrt{- \frac{2 \left(4 y^{2} + 18 y - 4\right)}{3 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}}} + 2 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}} + \frac{9}{4}}}{2} - \frac{\sqrt{\frac{2 \left(4 y^{2} + 18 y - 4\right)}{3 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}}} - 2 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}} + \frac{9}{2} + \frac{27}{4 \sqrt{- \frac{2 \left(4 y^{2} + 18 y - 4\right)}{3 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}}} + 2 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}} + \frac{9}{4}}}}}{2} - \frac{3}{4} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \frac{\sqrt{\frac{9}{4} - 2 \sqrt[3]{\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}}}}{2} - \frac{\sqrt{2 \sqrt[3]{\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}} + \frac{9}{2} + \frac{27}{4 \sqrt{\frac{9}{4} - 2 \sqrt[3]{\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}}}}}}{2} - \frac{3}{4} & \text{for}\: 4 y^{2} + 18 y - 4 = 0 \\- \frac{\sqrt{- \frac{2 \left(4 y^{2} + 18 y - 4\right)}{3 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}}} + 2 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}} + \frac{9}{4}}}{2} - \frac{\sqrt{\frac{2 \left(4 y^{2} + 18 y - 4\right)}{3 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}}} - 2 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}} + \frac{9}{2} + \frac{27}{4 \sqrt{- \frac{2 \left(4 y^{2} + 18 y - 4\right)}{3 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}}} + 2 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}} + \frac{9}{4}}}}}{2} - \frac{3}{4} & \text{otherwise} \end{cases}\right)}$$
// ___________________________________________________________________________ \ // ___________________________________________________________________________ \
|| / ___________________ | || / ___________________ |
|| / / 2 | || / / 2 |
|| / 9 / 9 9*y 81*y 27 | || / 9 / 9 9*y 81*y 27 |
|| / - + 2*3 / - - + ---- + ---- + ---------------------------------------- | || / - + 2*3 / - - + ---- + ---- + ---------------------------------------- |
|| / 2 \/ 2 2 4 ________________________________ ________________________________ | || / 2 \/ 2 2 4 ________________________________ ________________________________ |
|| / / ___________________ / ___________________ | || / / ___________________ / ___________________ |
|| / / / 2 / / 2 | || / / / 2 / / 2 |
|| / / 9 / 9 9*y 81*y / 9 / 9 9*y 81*y | || / / 9 / 9 9*y 81*y / 9 / 9 9*y 81*y |
|| / 4* / - - 2*3 / - - + ---- + ---- / - - 2*3 / - - + ---- + ---- | || / 4* / - - 2*3 / - - + ---- + ---- / - - 2*3 / - - + ---- + ---- |
|| 3 \/ \/ 4 \/ 2 2 4 \/ 4 \/ 2 2 4 2 | || 3 \/ \/ 4 \/ 2 2 4 \/ 4 \/ 2 2 4 2 |
|| - - + -------------------------------------------------------------------------------------- - -------------------------------------- for -4 + 4*y + 18*y = 0| || - - + -------------------------------------------------------------------------------------- - -------------------------------------- for -4 + 4*y + 18*y = 0|
|| 4 2 2 | || 4 2 2 |
|| | || |
|| _____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ | || _____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ |
|| / ________________________________________________________________________ | || / ________________________________________________________________________ |
|| / / ____________________________________________ | || / / ____________________________________________ |
|| / / / 2 | || / / / 2 |
|| / / / / 2 \ | || / / / / 2 \ |
|| / / / | 9 9*y 81*y| 3 | || / / / | 9 9*y 81*y| 3 |
|| / / / |- - + ---- + ----| / 2 \ 2 / 2 \ | || / / / |- - + ---- + ----| / 2 \ 2 / 2 \ |
x2 = I*im|< / 9 / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y 27 2*\-4 + 4*y + 18*y/ | + re|< / 9 / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y 27 2*\-4 + 4*y + 18*y/ |
|| / - - 2*3 / - + / -------------------- + ------------------- - ---- - ---- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------- | || / - - 2*3 / - + / -------------------- + ------------------- - ---- - ---- + --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------- |
|| / 2 \/ 4 \/ 4 27 8 4 _______________________________________________________________________________________________________________________________________________________________________________ ________________________________________________________________________ _______________________________________________________________________________________________________________________________________________________________________________ | || / 2 \/ 4 \/ 4 27 8 4 _______________________________________________________________________________________________________________________________________________________________________________ ________________________________________________________________________ _______________________________________________________________________________________________________________________________________________________________________________ |
|| / / ________________________________________________________________________ / ____________________________________________ / ________________________________________________________________________ | || / / ________________________________________________________________________ / ____________________________________________ / ________________________________________________________________________ |
|| / / / ____________________________________________ / / 2 / / ____________________________________________ | || / / / ____________________________________________ / / 2 / / ____________________________________________ |
|| / / / / 2 / / / 2 \ / / / 2 | || / / / / 2 / / / 2 \ / / / 2 |
|| / / / / / 2 \ / / | 9 9*y 81*y| 3 / / / / 2 \ | || / / / / / 2 \ / / | 9 9*y 81*y| 3 / / / / 2 \ |
|| / / / / | 9 9*y 81*y| 3 / / |- - + ---- + ----| / 2 \ 2 / / / | 9 9*y 81*y| 3 | || / / / / | 9 9*y 81*y| 3 / / |- - + ---- + ----| / 2 \ 2 / / / | 9 9*y 81*y| 3 |
|| / / / / |- - + ---- + ----| / 2 \ 2 / 2 \ / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y / / / |- - + ---- + ----| / 2 \ 2 / 2 \ | || / / / / |- - + ---- + ----| / 2 \ 2 / 2 \ / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y / / / |- - + ---- + ----| / 2 \ 2 / 2 \ |
|| / / 9 / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y 2*\-4 + 4*y + 18*y/ 3*3 / - + / -------------------- + ------------------- - ---- - ---- / 9 / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y 2*\-4 + 4*y + 18*y/ | || / / 9 / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y 2*\-4 + 4*y + 18*y/ 3*3 / - + / -------------------- + ------------------- - ---- - ---- / 9 / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y 2*\-4 + 4*y + 18*y/ |
|| / 4* / - + 2*3 / - + / -------------------- + ------------------- - ---- - ---- - ----------------------------------------------------------------------------------- \/ 4 \/ 4 27 8 4 / - + 2*3 / - + / -------------------- + ------------------- - ---- - ---- - ----------------------------------------------------------------------------------- | || / 4* / - + 2*3 / - + / -------------------- + ------------------- - ---- - ---- - ----------------------------------------------------------------------------------- \/ 4 \/ 4 27 8 4 / - + 2*3 / - + / -------------------- + ------------------- - ---- - ---- - ----------------------------------------------------------------------------------- |
|| / / 4 \/ 4 \/ 4 27 8 4 ________________________________________________________________________ / 4 \/ 4 \/ 4 27 8 4 ________________________________________________________________________ | || / / 4 \/ 4 \/ 4 27 8 4 ________________________________________________________________________ / 4 \/ 4 \/ 4 27 8 4 ________________________________________________________________________ |
|| / / / ____________________________________________ / / ____________________________________________ | || / / / ____________________________________________ / / ____________________________________________ |
|| / / / / 2 / / / 2 | || / / / / 2 / / / 2 |
|| / / / / / 2 \ / / / / 2 \ | || / / / / / 2 \ / / / / 2 \ |
|| / / / / | 9 9*y 81*y| 3 / / / | 9 9*y 81*y| 3 | || / / / / | 9 9*y 81*y| 3 / / / | 9 9*y 81*y| 3 |
|| / / / / |- - + ---- + ----| / 2 \ 2 / / / |- - + ---- + ----| / 2 \ 2 | || / / / / |- - + ---- + ----| / 2 \ 2 / / / |- - + ---- + ----| / 2 \ 2 |
|| / / / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y / / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y | || / / / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y / / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y |
|| / / 3*3 / - + / -------------------- + ------------------- - ---- - ---- / 3*3 / - + / -------------------- + ------------------- - ---- - ---- | || / / 3*3 / - + / -------------------- + ------------------- - ---- - ---- / 3*3 / - + / -------------------- + ------------------- - ---- - ---- |
|| 3 \/ \/ \/ 4 \/ 4 27 8 4 \/ \/ 4 \/ 4 27 8 4 | || 3 \/ \/ \/ 4 \/ 4 27 8 4 \/ \/ 4 \/ 4 27 8 4 |
||- - + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- otherwise | ||- - + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- otherwise |
\\ 4 2 2 / \\ 4 2 2 /
$$x_{2} = \operatorname{re}{\left(\begin{cases} - \frac{\sqrt{\frac{9}{4} - 2 \sqrt[3]{\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}}}}{2} + \frac{\sqrt{2 \sqrt[3]{\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}} + \frac{9}{2} + \frac{27}{4 \sqrt{\frac{9}{4} - 2 \sqrt[3]{\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}}}}}}{2} - \frac{3}{4} & \text{for}\: 4 y^{2} + 18 y - 4 = 0 \\- \frac{\sqrt{- \frac{2 \left(4 y^{2} + 18 y - 4\right)}{3 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}}} + 2 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}} + \frac{9}{4}}}{2} + \frac{\sqrt{\frac{2 \left(4 y^{2} + 18 y - 4\right)}{3 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}}} - 2 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}} + \frac{9}{2} + \frac{27}{4 \sqrt{- \frac{2 \left(4 y^{2} + 18 y - 4\right)}{3 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}}} + 2 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}} + \frac{9}{4}}}}}{2} - \frac{3}{4} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \frac{\sqrt{\frac{9}{4} - 2 \sqrt[3]{\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}}}}{2} + \frac{\sqrt{2 \sqrt[3]{\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}} + \frac{9}{2} + \frac{27}{4 \sqrt{\frac{9}{4} - 2 \sqrt[3]{\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}}}}}}{2} - \frac{3}{4} & \text{for}\: 4 y^{2} + 18 y - 4 = 0 \\- \frac{\sqrt{- \frac{2 \left(4 y^{2} + 18 y - 4\right)}{3 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}}} + 2 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}} + \frac{9}{4}}}{2} + \frac{\sqrt{\frac{2 \left(4 y^{2} + 18 y - 4\right)}{3 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}}} - 2 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}} + \frac{9}{2} + \frac{27}{4 \sqrt{- \frac{2 \left(4 y^{2} + 18 y - 4\right)}{3 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}}} + 2 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}} + \frac{9}{4}}}}}{2} - \frac{3}{4} & \text{otherwise} \end{cases}\right)}$$
// ___________________________________________________________________________ \ // ___________________________________________________________________________ \
|| / ___________________ | || / ___________________ |
|| / / 2 | || / / 2 |
|| / 9 / 9 9*y 81*y 27 | || / 9 / 9 9*y 81*y 27 |
|| / - + 2*3 / - - + ---- + ---- - ---------------------------------------- | || / - + 2*3 / - - + ---- + ---- - ---------------------------------------- |
|| ________________________________ / 2 \/ 2 2 4 ________________________________ | || ________________________________ / 2 \/ 2 2 4 ________________________________ |
|| / ___________________ / / ___________________ | || / ___________________ / / ___________________ |
|| / / 2 / / / 2 | || / / 2 / / / 2 |
|| / 9 / 9 9*y 81*y / / 9 / 9 9*y 81*y | || / 9 / 9 9*y 81*y / / 9 / 9 9*y 81*y |
|| / - - 2*3 / - - + ---- + ---- / 4* / - - 2*3 / - - + ---- + ---- | || / - - 2*3 / - - + ---- + ---- / 4* / - - 2*3 / - - + ---- + ---- |
|| 3 \/ 4 \/ 2 2 4 \/ \/ 4 \/ 2 2 4 2 | || 3 \/ 4 \/ 2 2 4 \/ \/ 4 \/ 2 2 4 2 |
|| - - + -------------------------------------- - -------------------------------------------------------------------------------------- for -4 + 4*y + 18*y = 0| || - - + -------------------------------------- - -------------------------------------------------------------------------------------- for -4 + 4*y + 18*y = 0|
|| 4 2 2 | || 4 2 2 |
|| | || |
|| _____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ | || _____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ |
|| / ________________________________________________________________________ | || / ________________________________________________________________________ |
|| / / ____________________________________________ | || / / ____________________________________________ |
|| / / / 2 | || / / / 2 |
|| / / / / 2 \ | || / / / / 2 \ |
|| / / / | 9 9*y 81*y| 3 | || / / / | 9 9*y 81*y| 3 |
|| / / / |- - + ---- + ----| / 2 \ 2 / 2 \ | || / / / |- - + ---- + ----| / 2 \ 2 / 2 \ |
x3 = I*im|< / 9 / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y 27 2*\-4 + 4*y + 18*y/ | + re|< / 9 / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y 27 2*\-4 + 4*y + 18*y/ |
|| / - - 2*3 / - + / -------------------- + ------------------- - ---- - ---- - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------- | || / - - 2*3 / - + / -------------------- + ------------------- - ---- - ---- - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------- |
|| _______________________________________________________________________________________________________________________________________________________________________________ / 2 \/ 4 \/ 4 27 8 4 _______________________________________________________________________________________________________________________________________________________________________________ ________________________________________________________________________ | || _______________________________________________________________________________________________________________________________________________________________________________ / 2 \/ 4 \/ 4 27 8 4 _______________________________________________________________________________________________________________________________________________________________________________ ________________________________________________________________________ |
|| / ________________________________________________________________________ / / ________________________________________________________________________ / ____________________________________________ | || / ________________________________________________________________________ / / ________________________________________________________________________ / ____________________________________________ |
|| / / ____________________________________________ / / / ____________________________________________ / / 2 | || / / ____________________________________________ / / / ____________________________________________ / / 2 |
|| / / / 2 / / / / 2 / / / 2 \ | || / / / 2 / / / / 2 / / / 2 \ |
|| / / / / 2 \ / / / / / 2 \ / / | 9 9*y 81*y| 3 | || / / / / 2 \ / / / / / 2 \ / / | 9 9*y 81*y| 3 |
|| / / / | 9 9*y 81*y| 3 / / / / | 9 9*y 81*y| 3 / / |- - + ---- + ----| / 2 \ 2 | || / / / | 9 9*y 81*y| 3 / / / / | 9 9*y 81*y| 3 / / |- - + ---- + ----| / 2 \ 2 |
|| / / / |- - + ---- + ----| / 2 \ 2 / 2 \ / / / / |- - + ---- + ----| / 2 \ 2 / 2 \ / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y | || / / / |- - + ---- + ----| / 2 \ 2 / 2 \ / / / / |- - + ---- + ----| / 2 \ 2 / 2 \ / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y |
|| / 9 / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y 2*\-4 + 4*y + 18*y/ / / 9 / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y 2*\-4 + 4*y + 18*y/ 3*3 / - + / -------------------- + ------------------- - ---- - ---- | || / 9 / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y 2*\-4 + 4*y + 18*y/ / / 9 / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y 2*\-4 + 4*y + 18*y/ 3*3 / - + / -------------------- + ------------------- - ---- - ---- |
|| / - + 2*3 / - + / -------------------- + ------------------- - ---- - ---- - ----------------------------------------------------------------------------------- / 4* / - + 2*3 / - + / -------------------- + ------------------- - ---- - ---- - ----------------------------------------------------------------------------------- \/ 4 \/ 4 27 8 4 | || / - + 2*3 / - + / -------------------- + ------------------- - ---- - ---- - ----------------------------------------------------------------------------------- / 4* / - + 2*3 / - + / -------------------- + ------------------- - ---- - ---- - ----------------------------------------------------------------------------------- \/ 4 \/ 4 27 8 4 |
|| / 4 \/ 4 \/ 4 27 8 4 ________________________________________________________________________ / / 4 \/ 4 \/ 4 27 8 4 ________________________________________________________________________ | || / 4 \/ 4 \/ 4 27 8 4 ________________________________________________________________________ / / 4 \/ 4 \/ 4 27 8 4 ________________________________________________________________________ |
|| / / ____________________________________________ / / / ____________________________________________ | || / / ____________________________________________ / / / ____________________________________________ |
|| / / / 2 / / / / 2 | || / / / 2 / / / / 2 |
|| / / / / 2 \ / / / / / 2 \ | || / / / / 2 \ / / / / / 2 \ |
|| / / / | 9 9*y 81*y| 3 / / / / | 9 9*y 81*y| 3 | || / / / | 9 9*y 81*y| 3 / / / / | 9 9*y 81*y| 3 |
|| / / / |- - + ---- + ----| / 2 \ 2 / / / / |- - + ---- + ----| / 2 \ 2 | || / / / |- - + ---- + ----| / 2 \ 2 / / / / |- - + ---- + ----| / 2 \ 2 |
|| / / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y / / / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y | || / / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y / / / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y |
|| / 3*3 / - + / -------------------- + ------------------- - ---- - ---- / / 3*3 / - + / -------------------- + ------------------- - ---- - ---- | || / 3*3 / - + / -------------------- + ------------------- - ---- - ---- / / 3*3 / - + / -------------------- + ------------------- - ---- - ---- |
|| 3 \/ \/ 4 \/ 4 27 8 4 \/ \/ \/ 4 \/ 4 27 8 4 | || 3 \/ \/ 4 \/ 4 27 8 4 \/ \/ \/ 4 \/ 4 27 8 4 |
||- - + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- otherwise | ||- - + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- otherwise |
\\ 4 2 2 / \\ 4 2 2 /
$$x_{3} = \operatorname{re}{\left(\begin{cases} \frac{\sqrt{\frac{9}{4} - 2 \sqrt[3]{\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}}}}{2} - \frac{\sqrt{2 \sqrt[3]{\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}} + \frac{9}{2} - \frac{27}{4 \sqrt{\frac{9}{4} - 2 \sqrt[3]{\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}}}}}}{2} - \frac{3}{4} & \text{for}\: 4 y^{2} + 18 y - 4 = 0 \\\frac{\sqrt{- \frac{2 \left(4 y^{2} + 18 y - 4\right)}{3 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}}} + 2 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}} + \frac{9}{4}}}{2} - \frac{\sqrt{\frac{2 \left(4 y^{2} + 18 y - 4\right)}{3 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}}} - 2 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}} + \frac{9}{2} - \frac{27}{4 \sqrt{- \frac{2 \left(4 y^{2} + 18 y - 4\right)}{3 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}}} + 2 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}} + \frac{9}{4}}}}}{2} - \frac{3}{4} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{\sqrt{\frac{9}{4} - 2 \sqrt[3]{\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}}}}{2} - \frac{\sqrt{2 \sqrt[3]{\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}} + \frac{9}{2} - \frac{27}{4 \sqrt{\frac{9}{4} - 2 \sqrt[3]{\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}}}}}}{2} - \frac{3}{4} & \text{for}\: 4 y^{2} + 18 y - 4 = 0 \\\frac{\sqrt{- \frac{2 \left(4 y^{2} + 18 y - 4\right)}{3 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}}} + 2 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}} + \frac{9}{4}}}{2} - \frac{\sqrt{\frac{2 \left(4 y^{2} + 18 y - 4\right)}{3 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}}} - 2 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}} + \frac{9}{2} - \frac{27}{4 \sqrt{- \frac{2 \left(4 y^{2} + 18 y - 4\right)}{3 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}}} + 2 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}} + \frac{9}{4}}}}}{2} - \frac{3}{4} & \text{otherwise} \end{cases}\right)}$$
// ___________________________________________________________________________ \ // ___________________________________________________________________________ \
|| / ___________________ | || / ___________________ |
|| / / 2 | || / / 2 |
|| / 9 / 9 9*y 81*y 27 | || / 9 / 9 9*y 81*y 27 |
|| / - + 2*3 / - - + ---- + ---- - ---------------------------------------- | || / - + 2*3 / - - + ---- + ---- - ---------------------------------------- |
|| ________________________________ / 2 \/ 2 2 4 ________________________________ | || ________________________________ / 2 \/ 2 2 4 ________________________________ |
|| / ___________________ / / ___________________ | || / ___________________ / / ___________________ |
|| / / 2 / / / 2 | || / / 2 / / / 2 |
|| / 9 / 9 9*y 81*y / / 9 / 9 9*y 81*y | || / 9 / 9 9*y 81*y / / 9 / 9 9*y 81*y |
|| / - - 2*3 / - - + ---- + ---- / 4* / - - 2*3 / - - + ---- + ---- | || / - - 2*3 / - - + ---- + ---- / 4* / - - 2*3 / - - + ---- + ---- |
|| 3 \/ 4 \/ 2 2 4 \/ \/ 4 \/ 2 2 4 2 | || 3 \/ 4 \/ 2 2 4 \/ \/ 4 \/ 2 2 4 2 |
|| - - + -------------------------------------- + -------------------------------------------------------------------------------------- for -4 + 4*y + 18*y = 0| || - - + -------------------------------------- + -------------------------------------------------------------------------------------- for -4 + 4*y + 18*y = 0|
|| 4 2 2 | || 4 2 2 |
|| | || |
|| _____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ | || _____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ |
|| / ________________________________________________________________________ | || / ________________________________________________________________________ |
|| / / ____________________________________________ | || / / ____________________________________________ |
|| / / / 2 | || / / / 2 |
|| / / / / 2 \ | || / / / / 2 \ |
|| / / / | 9 9*y 81*y| 3 | || / / / | 9 9*y 81*y| 3 |
|| / / / |- - + ---- + ----| / 2 \ 2 / 2 \ | || / / / |- - + ---- + ----| / 2 \ 2 / 2 \ |
x4 = I*im|< / 9 / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y 27 2*\-4 + 4*y + 18*y/ | + re|< / 9 / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y 27 2*\-4 + 4*y + 18*y/ |
|| / - - 2*3 / - + / -------------------- + ------------------- - ---- - ---- - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------- | || / - - 2*3 / - + / -------------------- + ------------------- - ---- - ---- - --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------- |
|| _______________________________________________________________________________________________________________________________________________________________________________ / 2 \/ 4 \/ 4 27 8 4 _______________________________________________________________________________________________________________________________________________________________________________ ________________________________________________________________________ | || _______________________________________________________________________________________________________________________________________________________________________________ / 2 \/ 4 \/ 4 27 8 4 _______________________________________________________________________________________________________________________________________________________________________________ ________________________________________________________________________ |
|| / ________________________________________________________________________ / / ________________________________________________________________________ / ____________________________________________ | || / ________________________________________________________________________ / / ________________________________________________________________________ / ____________________________________________ |
|| / / ____________________________________________ / / / ____________________________________________ / / 2 | || / / ____________________________________________ / / / ____________________________________________ / / 2 |
|| / / / 2 / / / / 2 / / / 2 \ | || / / / 2 / / / / 2 / / / 2 \ |
|| / / / / 2 \ / / / / / 2 \ / / | 9 9*y 81*y| 3 | || / / / / 2 \ / / / / / 2 \ / / | 9 9*y 81*y| 3 |
|| / / / | 9 9*y 81*y| 3 / / / / | 9 9*y 81*y| 3 / / |- - + ---- + ----| / 2 \ 2 | || / / / | 9 9*y 81*y| 3 / / / / | 9 9*y 81*y| 3 / / |- - + ---- + ----| / 2 \ 2 |
|| / / / |- - + ---- + ----| / 2 \ 2 / 2 \ / / / / |- - + ---- + ----| / 2 \ 2 / 2 \ / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y | || / / / |- - + ---- + ----| / 2 \ 2 / 2 \ / / / / |- - + ---- + ----| / 2 \ 2 / 2 \ / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y |
|| / 9 / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y 2*\-4 + 4*y + 18*y/ / / 9 / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y 2*\-4 + 4*y + 18*y/ 3*3 / - + / -------------------- + ------------------- - ---- - ---- | || / 9 / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y 2*\-4 + 4*y + 18*y/ / / 9 / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y 2*\-4 + 4*y + 18*y/ 3*3 / - + / -------------------- + ------------------- - ---- - ---- |
|| / - + 2*3 / - + / -------------------- + ------------------- - ---- - ---- - ----------------------------------------------------------------------------------- / 4* / - + 2*3 / - + / -------------------- + ------------------- - ---- - ---- - ----------------------------------------------------------------------------------- \/ 4 \/ 4 27 8 4 | || / - + 2*3 / - + / -------------------- + ------------------- - ---- - ---- - ----------------------------------------------------------------------------------- / 4* / - + 2*3 / - + / -------------------- + ------------------- - ---- - ---- - ----------------------------------------------------------------------------------- \/ 4 \/ 4 27 8 4 |
|| / 4 \/ 4 \/ 4 27 8 4 ________________________________________________________________________ / / 4 \/ 4 \/ 4 27 8 4 ________________________________________________________________________ | || / 4 \/ 4 \/ 4 27 8 4 ________________________________________________________________________ / / 4 \/ 4 \/ 4 27 8 4 ________________________________________________________________________ |
|| / / ____________________________________________ / / / ____________________________________________ | || / / ____________________________________________ / / / ____________________________________________ |
|| / / / 2 / / / / 2 | || / / / 2 / / / / 2 |
|| / / / / 2 \ / / / / / 2 \ | || / / / / 2 \ / / / / / 2 \ |
|| / / / | 9 9*y 81*y| 3 / / / / | 9 9*y 81*y| 3 | || / / / | 9 9*y 81*y| 3 / / / / | 9 9*y 81*y| 3 |
|| / / / |- - + ---- + ----| / 2 \ 2 / / / / |- - + ---- + ----| / 2 \ 2 | || / / / |- - + ---- + ----| / 2 \ 2 / / / / |- - + ---- + ----| / 2 \ 2 |
|| / / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y / / / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y | || / / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y / / / 9 / \ 2 2 4 / \-4 + 4*y + 18*y/ 81*y 9*y |
|| / 3*3 / - + / -------------------- + ------------------- - ---- - ---- / / 3*3 / - + / -------------------- + ------------------- - ---- - ---- | || / 3*3 / - + / -------------------- + ------------------- - ---- - ---- / / 3*3 / - + / -------------------- + ------------------- - ---- - ---- |
|| 3 \/ \/ 4 \/ 4 27 8 4 \/ \/ \/ 4 \/ 4 27 8 4 | || 3 \/ \/ 4 \/ 4 27 8 4 \/ \/ \/ 4 \/ 4 27 8 4 |
||- - + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- otherwise | ||- - + ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- otherwise |
\\ 4 2 2 / \\ 4 2 2 /
$$x_{4} = \operatorname{re}{\left(\begin{cases} \frac{\sqrt{\frac{9}{4} - 2 \sqrt[3]{\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}}}}{2} + \frac{\sqrt{2 \sqrt[3]{\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}} + \frac{9}{2} - \frac{27}{4 \sqrt{\frac{9}{4} - 2 \sqrt[3]{\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}}}}}}{2} - \frac{3}{4} & \text{for}\: 4 y^{2} + 18 y - 4 = 0 \\\frac{\sqrt{- \frac{2 \left(4 y^{2} + 18 y - 4\right)}{3 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}}} + 2 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}} + \frac{9}{4}}}{2} + \frac{\sqrt{\frac{2 \left(4 y^{2} + 18 y - 4\right)}{3 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}}} - 2 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}} + \frac{9}{2} - \frac{27}{4 \sqrt{- \frac{2 \left(4 y^{2} + 18 y - 4\right)}{3 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}}} + 2 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}} + \frac{9}{4}}}}}{2} - \frac{3}{4} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{\sqrt{\frac{9}{4} - 2 \sqrt[3]{\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}}}}{2} + \frac{\sqrt{2 \sqrt[3]{\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}} + \frac{9}{2} - \frac{27}{4 \sqrt{\frac{9}{4} - 2 \sqrt[3]{\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}}}}}}{2} - \frac{3}{4} & \text{for}\: 4 y^{2} + 18 y - 4 = 0 \\\frac{\sqrt{- \frac{2 \left(4 y^{2} + 18 y - 4\right)}{3 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}}} + 2 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}} + \frac{9}{4}}}{2} + \frac{\sqrt{\frac{2 \left(4 y^{2} + 18 y - 4\right)}{3 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}}} - 2 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}} + \frac{9}{2} - \frac{27}{4 \sqrt{- \frac{2 \left(4 y^{2} + 18 y - 4\right)}{3 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}}} + 2 \sqrt[3]{- \frac{9 y^{2}}{4} - \frac{81 y}{8} + \sqrt{\frac{\left(4 y^{2} + 18 y - 4\right)^{3}}{27} + \frac{\left(\frac{9 y^{2}}{2} + \frac{81 y}{4} - \frac{9}{2}\right)^{2}}{4}} + \frac{9}{4}} + \frac{9}{4}}}}}{2} - \frac{3}{4} & \text{otherwise} \end{cases}\right)}$$
x4 = re(Piecewise((sqrt(9/4 - 2*(9*y^2/2 + 81*y/4 - 9/2)^(1/3))/2 + sqrt(2*(9*y^2/2 + 81*y/4 - 9/2)^(1/3) + 9/2 - 27/(4*sqrt(9/4 - 2*(9*y^2/2 + 81*y/4 - 9/2)^(1/3)))/2 - 3/4, 4*y^2 + 18*y - 4 = 0), (sqrt(-2*(4*y^2 + 18*y - 4)/(3*(-9*y^2/4 - 81*y/8 + sqrt((4*y^2 + 18*y - 4)^3/27 + (9*y^2/2 + 81*y/4 - 9/2)^2/4) + 9/4)^(1/3)) + 2*(-9*y^2/4 - 81*y/8 + sqrt((4*y^2 + 18*y - 4)^3/27 + (9*y^2/2 + 81*y/4 - 9/2)^2/4) + 9/4)^(1/3) + 9/4)/2 + sqrt(2*(4*y^2 + 18*y - 4)/(3*(-9*y^2/4 - 81*y/8 + sqrt((4*y^2 + 18*y - 4)^3/27 + (9*y^2/2 + 81*y/4 - 9/2)^2/4) + 9/4)^(1/3)) - 2*(-9*y^2/4 - 81*y/8 + sqrt((4*y^2 + 18*y - 4)^3/27 + (9*y^2/2 + 81*y/4 - 9/2)^2/4) + 9/4)^(1/3) + 9/2 - 27/(4*sqrt(-2*(4*y^2 + 18*y - 4)/(3*(-9*y^2/4 - 81*y/8 + sqrt((4*y^2 + 18*y - 4)^3/27 + (9*y^2/2 + 81*y/4 - 9/2)^2/4) + 9/4)^(1/3)) + 2*(-9*y^2/4 - 81*y/8 + sqrt((4*y^2 + 18*y - 4)^3/27 + (9*y^2/2 + 81*y/4 - 9/2)^2/4) + 9/4)^(1/3) + 9/4)))/2 - 3/4, True))) + i*im(Piecewise((sqrt(9/4 - 2*(9*y^2/2 + 81*y/4 - 9/2)^(1/3))/2 + sqrt(2*(9*y^2/2 + 81*y/4 - 9/2)^(1/3) + 9/2 - 27/(4*sqrt(9/4 - 2*(9*y^2/2 + 81*y/4 - 9/2)^(1/3))))/2 - 3/4, 4*y^2 + 18*y - 4 = 0), (sqrt(-2*(4*y^2 + 18*y - 4)/(3*(-9*y^2/4 - 81*y/8 + sqrt((4*y^2 + 18*y - 4)^3/27 + (9*y^2/2 + 81*y/4 - 9/2)^2/4) + 9/4)^(1/3)) + 2*(-9*y^2/4 - 81*y/8 + sqrt((4*y^2 + 18*y - 4)^3/27 + (9*y^2/2 + 81*y/4 - 9/2)^2/4) + 9/4)^(1/3) + 9/4)/2 + sqrt(2*(4*y^2 + 18*y - 4)/(3*(-9*y^2/4 - 81*y/8 + sqrt((4*y^2 + 18*y - 4)^3/27 + (9*y^2/2 + 81*y/4 - 9/2)^2/4) + 9/4)^(1/3)) - 2*(-9*y^2/4 - 81*y/8 + sqrt((4*y^2 + 18*y - 4)^3/27 + (9*y^2/2 + 81*y/4 - 9/2)^2/4) + 9/4)^(1/3) + 9/2 - 27/(4*sqrt(-2*(4*y^2 + 18*y - 4)/(3*(-9*y^2/4 - 81*y/8 + sqrt((4*y^2 + 18*y - 4)^3/27 + (9*y^2/2 + 81*y/4 - 9/2)^2/4) + 9/4)^(1/3)) + 2*(-9*y^2/4 - 81*y/8 + sqrt((4*y^2 + 18*y - 4)^3/27 + (9*y^2/2 + 81*y/4 - 9/2)^2/4) + 9/4)^(1/3) + 9/4)))/2 - 3/4, True))))