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2*x^3*y+(-6)*x*y-3*y=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Teorema de Cardano-Vieta
reescribamos la ecuación
$$- 3 y + \left(- 6 x y + 2 x^{3} y\right) = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$\frac{2 x^{3} y - 6 x y - 3 y}{2 y} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -3$$
$$v = \frac{d}{a}$$
$$v = - \frac{3}{2}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -3$$
$$x_{1} x_{2} x_{3} = - \frac{3}{2}$$
$$- 3 y + \left(- 6 x y + 2 x^{3} y\right) = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$\frac{2 x^{3} y - 6 x y - 3 y}{2 y} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -3$$
$$v = \frac{d}{a}$$
$$v = - \frac{3}{2}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -3$$
$$x_{1} x_{2} x_{3} = - \frac{3}{2}$$
Gráfica
Respuesta rápida
[src]
/ / ___\\ / / / ___\\ / / ___\\ \ / / ___\\
| |\/ 7 || | | |\/ 7 || | |\/ 7 || | | |\/ 7 ||
|atan|-----|| | |atan|-----|| |atan|-----|| | |atan|-----||
| \ 3 /| | | \ 3 /| ___ | \ 3 /| | ___ | \ 3 /|
cos|-----------| | sin|-----------| \/ 3 *cos|-----------| | \/ 3 *sin|-----------|
\ 3 / | \ 3 / \ 3 / / 1 \| \ 3 / / 1 \
x1 = - ---------------- + I*|- ---------------- - ---------------------- + im|----------------------------------|| + ---------------------- + re|----------------------------------|
2 | 2 2 | _____________|| 2 | _____________|
| |/ ___\ / ___ || |/ ___\ / ___ |
| || 1 I*\/ 3 | / 3 I*\/ 7 || || 1 I*\/ 3 | / 3 I*\/ 7 |
| ||- - - -------|*3 / - + ------- || ||- - - -------|*3 / - + ------- |
\ \\ 2 2 / \/ 4 4 // \\ 2 2 / \/ 4 4 /
$$x_{1} = - \frac{\cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)} + \frac{\sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} - \frac{\sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)}\right)$$
/ / ___\\ / / / ___\\ / / ___\\ \ / / ___\\
| |\/ 7 || | | |\/ 7 || | |\/ 7 || | | |\/ 7 ||
|atan|-----|| | |atan|-----|| |atan|-----|| | |atan|-----||
| \ 3 /| | | \ 3 /| ___ | \ 3 /| | ___ | \ 3 /|
cos|-----------| | sin|-----------| \/ 3 *cos|-----------| | \/ 3 *sin|-----------|
\ 3 / | \ 3 / \ 3 / / 1 \| \ 3 / / 1 \
x2 = - ---------------- + I*|- ---------------- + ---------------------- + im|----------------------------------|| - ---------------------- + re|----------------------------------|
2 | 2 2 | _____________|| 2 | _____________|
| |/ ___\ / ___ || |/ ___\ / ___ |
| || 1 I*\/ 3 | / 3 I*\/ 7 || || 1 I*\/ 3 | / 3 I*\/ 7 |
| ||- - + -------|*3 / - + ------- || ||- - + -------|*3 / - + ------- |
\ \\ 2 2 / \/ 4 4 // \\ 2 2 / \/ 4 4 /
$$x_{2} = \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)} - \frac{\cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} - \frac{\sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + i \left(\operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)} - \frac{\sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2}\right)$$
_____________
/ ___
1 / 3 I*\/ 7
x3 = ------------------ + 3 / - + -------
_____________ \/ 4 4
/ ___
/ 3 I*\/ 7
3 / - + -------
\/ 4 4
$$x_{3} = \frac{1}{\sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}} + \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}$$
x3 = (3/4 + sqrt(7)*i/4)^(-1/3) + (3/4 + sqrt(7)*i/4)^(1/3)
Suma y producto de raíces
[src]
suma
/ / ___\\ / / / ___\\ / / ___\\ \ / / ___\\ / / ___\\ / / / ___\\ / / ___\\ \ / / ___\\
| |\/ 7 || | | |\/ 7 || | |\/ 7 || | | |\/ 7 || | |\/ 7 || | | |\/ 7 || | |\/ 7 || | | |\/ 7 ||
|atan|-----|| | |atan|-----|| |atan|-----|| | |atan|-----|| |atan|-----|| | |atan|-----|| |atan|-----|| | |atan|-----||
| \ 3 /| | | \ 3 /| ___ | \ 3 /| | ___ | \ 3 /| | \ 3 /| | | \ 3 /| ___ | \ 3 /| | ___ | \ 3 /| _____________
cos|-----------| | sin|-----------| \/ 3 *cos|-----------| | \/ 3 *sin|-----------| cos|-----------| | sin|-----------| \/ 3 *cos|-----------| | \/ 3 *sin|-----------| / ___
\ 3 / | \ 3 / \ 3 / / 1 \| \ 3 / / 1 \ \ 3 / | \ 3 / \ 3 / / 1 \| \ 3 / / 1 \ 1 / 3 I*\/ 7
- ---------------- + I*|- ---------------- - ---------------------- + im|----------------------------------|| + ---------------------- + re|----------------------------------| + - ---------------- + I*|- ---------------- + ---------------------- + im|----------------------------------|| - ---------------------- + re|----------------------------------| + ------------------ + 3 / - + -------
2 | 2 2 | _____________|| 2 | _____________| 2 | 2 2 | _____________|| 2 | _____________| _____________ \/ 4 4
| |/ ___\ / ___ || |/ ___\ / ___ | | |/ ___\ / ___ || |/ ___\ / ___ | / ___
| || 1 I*\/ 3 | / 3 I*\/ 7 || || 1 I*\/ 3 | / 3 I*\/ 7 | | || 1 I*\/ 3 | / 3 I*\/ 7 || || 1 I*\/ 3 | / 3 I*\/ 7 | / 3 I*\/ 7
| ||- - - -------|*3 / - + ------- || ||- - - -------|*3 / - + ------- | | ||- - + -------|*3 / - + ------- || ||- - + -------|*3 / - + ------- | 3 / - + -------
\ \\ 2 2 / \/ 4 4 // \\ 2 2 / \/ 4 4 / \ \\ 2 2 / \/ 4 4 // \\ 2 2 / \/ 4 4 / \/ 4 4
$$\left(\left(\operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)} - \frac{\cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} - \frac{\sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + i \left(\operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)} - \frac{\sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2}\right)\right) + \left(- \frac{\cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)} + \frac{\sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} - \frac{\sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)}\right)\right)\right) + \left(\frac{1}{\sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}} + \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}\right)$$
=
/ / / ___\\ / / ___\\ \ / / / ___\\ / / ___\\ \
| | |\/ 7 || | |\/ 7 || | | | |\/ 7 || | |\/ 7 || |
/ / ___\\ | |atan|-----|| |atan|-----|| | | |atan|-----|| |atan|-----|| |
_____________ | |\/ 7 || | | \ 3 /| ___ | \ 3 /| | | | \ 3 /| ___ | \ 3 /| |
/ ___ |atan|-----|| | sin|-----------| \/ 3 *cos|-----------| | | sin|-----------| \/ 3 *cos|-----------| |
1 / 3 I*\/ 7 | \ 3 /| | \ 3 / \ 3 / / 1 \| | \ 3 / \ 3 / / 1 \| / 1 \ / 1 \
------------------ + 3 / - + ------- - cos|-----------| + I*|- ---------------- + ---------------------- + im|----------------------------------|| + I*|- ---------------- - ---------------------- + im|----------------------------------|| + re|----------------------------------| + re|----------------------------------|
_____________ \/ 4 4 \ 3 / | 2 2 | _____________|| | 2 2 | _____________|| | _____________| | _____________|
/ ___ | |/ ___\ / ___ || | |/ ___\ / ___ || |/ ___\ / ___ | |/ ___\ / ___ |
/ 3 I*\/ 7 | || 1 I*\/ 3 | / 3 I*\/ 7 || | || 1 I*\/ 3 | / 3 I*\/ 7 || || 1 I*\/ 3 | / 3 I*\/ 7 | || 1 I*\/ 3 | / 3 I*\/ 7 |
3 / - + ------- | ||- - + -------|*3 / - + ------- || | ||- - - -------|*3 / - + ------- || ||- - + -------|*3 / - + ------- | ||- - - -------|*3 / - + ------- |
\/ 4 4 \ \\ 2 2 / \/ 4 4 // \ \\ 2 2 / \/ 4 4 // \\ 2 2 / \/ 4 4 / \\ 2 2 / \/ 4 4 /
$$- \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)} + \frac{1}{\sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}} + i \left(\operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)} - \frac{\sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2}\right) + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} - \frac{\sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)}\right) + \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}$$
producto
/ / / ___\\ / / / ___\\ / / ___\\ \ / / ___\\ \ / / / ___\\ / / / ___\\ / / ___\\ \ / / ___\\ \ | | |\/ 7 || | | |\/ 7 || | |\/ 7 || | | |\/ 7 || | | | |\/ 7 || | | |\/ 7 || | |\/ 7 || | | |\/ 7 || | | |atan|-----|| | |atan|-----|| |atan|-----|| | |atan|-----|| | | |atan|-----|| | |atan|-----|| |atan|-----|| | |atan|-----|| | | | \ 3 /| | | \ 3 /| ___ | \ 3 /| | ___ | \ 3 /| | | | \ 3 /| | | \ 3 /| ___ | \ 3 /| | ___ | \ 3 /| | / _____________\ | cos|-----------| | sin|-----------| \/ 3 *cos|-----------| | \/ 3 *sin|-----------| | | cos|-----------| | sin|-----------| \/ 3 *cos|-----------| | \/ 3 *sin|-----------| | | / ___ | | \ 3 / | \ 3 / \ 3 / / 1 \| \ 3 / / 1 \| | \ 3 / | \ 3 / \ 3 / / 1 \| \ 3 / / 1 \| | 1 / 3 I*\/ 7 | |- ---------------- + I*|- ---------------- - ---------------------- + im|----------------------------------|| + ---------------------- + re|----------------------------------||*|- ---------------- + I*|- ---------------- + ---------------------- + im|----------------------------------|| - ---------------------- + re|----------------------------------||*|------------------ + 3 / - + ------- | | 2 | 2 2 | _____________|| 2 | _____________|| | 2 | 2 2 | _____________|| 2 | _____________|| | _____________ \/ 4 4 | | | |/ ___\ / ___ || |/ ___\ / ___ || | | |/ ___\ / ___ || |/ ___\ / ___ || | / ___ | | | || 1 I*\/ 3 | / 3 I*\/ 7 || || 1 I*\/ 3 | / 3 I*\/ 7 || | | || 1 I*\/ 3 | / 3 I*\/ 7 || || 1 I*\/ 3 | / 3 I*\/ 7 || | / 3 I*\/ 7 | | | ||- - - -------|*3 / - + ------- || ||- - - -------|*3 / - + ------- || | | ||- - + -------|*3 / - + ------- || ||- - + -------|*3 / - + ------- || |3 / - + ------- | \ \ \\ 2 2 / \/ 4 4 // \\ 2 2 / \/ 4 4 // \ \ \\ 2 2 / \/ 4 4 // \\ 2 2 / \/ 4 4 // \\/ 4 4 /
$$\left(- \frac{\cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)} + \frac{\sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} - \frac{\sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)}\right)\right) \left(\operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)} - \frac{\cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} - \frac{\sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + i \left(\operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)} - \frac{\sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2}\right)\right) \left(\frac{1}{\sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}} + \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}\right)$$
=
/ / / ___\\\
| | |\/ 7 |||
| |2*atan|-----||| / 2/3 \
3 ___ | | \ 3 /|| |/ ___\ 3 ___|
\/ 2 *|-1 + 2*cos|-------------||*\\3 + I*\/ 7 / + 2*\/ 2 /
\ \ 3 //
--------------------------------------------------------------
_____________
3 / ___
2*\/ 3 + I*\/ 7
$$\frac{\sqrt[3]{2} \left(-1 + 2 \cos{\left(\frac{2 \operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}\right) \left(2 \sqrt[3]{2} + \left(3 + \sqrt{7} i\right)^{\frac{2}{3}}\right)}{2 \sqrt[3]{3 + \sqrt{7} i}}$$
2^(1/3)*(-1 + 2*cos(2*atan(sqrt(7)/3)/3))*((3 + i*sqrt(7))^(2/3) + 2*2^(1/3))/(2*(3 + i*sqrt(7))^(1/3))