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X^3+0,2x+2=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = \frac{1}{5}$$
$$v = \frac{d}{a}$$
$$v = 2$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = \frac{1}{5}$$
$$x_{1} x_{2} x_{3} = 2$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = \frac{1}{5}$$
$$v = \frac{d}{a}$$
$$v = 2$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = \frac{1}{5}$$
$$x_{1} x_{2} x_{3} = 2$$
Respuesta rápida
[src]
__________________ / __________________ \
/ ______ | / ______ |
/ 12*\/ 3165 | ___ / 12*\/ 3165 |
3 / 27 + ----------- |\/ 3 *3 / 27 + ----------- ___ |
1 \/ 25 | \/ 25 \/ 3 |
x1 = - -------------------------- + ----------------------- + I*|----------------------------- + --------------------------|
__________________ 6 | 6 __________________|
/ ______ | / ______ |
/ 12*\/ 3165 | / 12*\/ 3165 |
10*3 / 27 + ----------- | 10*3 / 27 + ----------- |
\/ 25 \ \/ 25 /
$$x_{1} = - \frac{1}{10 \sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}} + \frac{\sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}}{6} + i \left(\frac{\sqrt{3}}{10 \sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}} + \frac{\sqrt{3} \sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}}{6}\right)$$
__________________ / __________________ \
/ ______ | / ______ |
/ 12*\/ 3165 | ___ / 12*\/ 3165 |
3 / 27 + ----------- | \/ 3 *3 / 27 + ----------- ___ |
1 \/ 25 | \/ 25 \/ 3 |
x2 = - -------------------------- + ----------------------- + I*|- ----------------------------- - --------------------------|
__________________ 6 | 6 __________________|
/ ______ | / ______ |
/ 12*\/ 3165 | / 12*\/ 3165 |
10*3 / 27 + ----------- | 10*3 / 27 + ----------- |
\/ 25 \ \/ 25 /
$$x_{2} = - \frac{1}{10 \sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}} + \frac{\sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}}{6} - \frac{\sqrt{3}}{10 \sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}}\right)$$
__________________
/ ______
/ 12*\/ 3165
3 / 27 + -----------
\/ 25 1
x3 = - ----------------------- + -------------------------
3 __________________
/ ______
/ 12*\/ 3165
5*3 / 27 + -----------
\/ 25
$$x_{3} = - \frac{\sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}}{3} + \frac{1}{5 \sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}}$$
x3 = -(27 + 12*sqrt(3165)/25)^(1/3)/3 + 1/(5*(27 + 12*sqrt(3165)/25)^(1/3))
Suma y producto de raíces
[src]
suma
__________________ / __________________ \ __________________ / __________________ \ __________________
/ ______ | / ______ | / ______ | / ______ | / ______
/ 12*\/ 3165 | ___ / 12*\/ 3165 | / 12*\/ 3165 | ___ / 12*\/ 3165 | / 12*\/ 3165
3 / 27 + ----------- |\/ 3 *3 / 27 + ----------- ___ | 3 / 27 + ----------- | \/ 3 *3 / 27 + ----------- ___ | 3 / 27 + -----------
1 \/ 25 | \/ 25 \/ 3 | 1 \/ 25 | \/ 25 \/ 3 | \/ 25 1
- -------------------------- + ----------------------- + I*|----------------------------- + --------------------------| + - -------------------------- + ----------------------- + I*|- ----------------------------- - --------------------------| + - ----------------------- + -------------------------
__________________ 6 | 6 __________________| __________________ 6 | 6 __________________| 3 __________________
/ ______ | / ______ | / ______ | / ______ | / ______
/ 12*\/ 3165 | / 12*\/ 3165 | / 12*\/ 3165 | / 12*\/ 3165 | / 12*\/ 3165
10*3 / 27 + ----------- | 10*3 / 27 + ----------- | 10*3 / 27 + ----------- | 10*3 / 27 + ----------- | 5*3 / 27 + -----------
\/ 25 \ \/ 25 / \/ 25 \ \/ 25 / \/ 25
$$\left(- \frac{\sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}}{3} + \frac{1}{5 \sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}}\right) + \left(\left(- \frac{1}{10 \sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}} + \frac{\sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}}{6} - \frac{\sqrt{3}}{10 \sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}}\right)\right) + \left(- \frac{1}{10 \sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}} + \frac{\sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}}{6} + i \left(\frac{\sqrt{3}}{10 \sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}} + \frac{\sqrt{3} \sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}}{6}\right)\right)\right)$$
=
/ __________________ \ / __________________ \ | / ______ | | / ______ | | ___ / 12*\/ 3165 | | ___ / 12*\/ 3165 | | \/ 3 *3 / 27 + ----------- ___ | |\/ 3 *3 / 27 + ----------- ___ | | \/ 25 \/ 3 | | \/ 25 \/ 3 | I*|- ----------------------------- - --------------------------| + I*|----------------------------- + --------------------------| | 6 __________________| | 6 __________________| | / ______ | | / ______ | | / 12*\/ 3165 | | / 12*\/ 3165 | | 10*3 / 27 + ----------- | | 10*3 / 27 + ----------- | \ \/ 25 / \ \/ 25 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}}{6} - \frac{\sqrt{3}}{10 \sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}}\right) + i \left(\frac{\sqrt{3}}{10 \sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}} + \frac{\sqrt{3} \sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}}{6}\right)$$
producto
/ __________________ / __________________ \\ / __________________ / __________________ \\ / __________________ \ | / ______ | / ______ || | / ______ | / ______ || | / ______ | | / 12*\/ 3165 | ___ / 12*\/ 3165 || | / 12*\/ 3165 | ___ / 12*\/ 3165 || | / 12*\/ 3165 | | 3 / 27 + ----------- |\/ 3 *3 / 27 + ----------- ___ || | 3 / 27 + ----------- | \/ 3 *3 / 27 + ----------- ___ || | 3 / 27 + ----------- | | 1 \/ 25 | \/ 25 \/ 3 || | 1 \/ 25 | \/ 25 \/ 3 || | \/ 25 1 | |- -------------------------- + ----------------------- + I*|----------------------------- + --------------------------||*|- -------------------------- + ----------------------- + I*|- ----------------------------- - --------------------------||*|- ----------------------- + -------------------------| | __________________ 6 | 6 __________________|| | __________________ 6 | 6 __________________|| | 3 __________________| | / ______ | / ______ || | / ______ | / ______ || | / ______ | | / 12*\/ 3165 | / 12*\/ 3165 || | / 12*\/ 3165 | / 12*\/ 3165 || | / 12*\/ 3165 | | 10*3 / 27 + ----------- | 10*3 / 27 + ----------- || | 10*3 / 27 + ----------- | 10*3 / 27 + ----------- || | 5*3 / 27 + ----------- | \ \/ 25 \ \/ 25 // \ \/ 25 \ \/ 25 // \ \/ 25 /
$$\left(- \frac{1}{10 \sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}} + \frac{\sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}}{6} + i \left(\frac{\sqrt{3}}{10 \sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}} + \frac{\sqrt{3} \sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}}{6}\right)\right) \left(- \frac{1}{10 \sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}} + \frac{\sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}}{6} - \frac{\sqrt{3}}{10 \sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}}\right)\right) \left(- \frac{\sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}}{3} + \frac{1}{5 \sqrt[3]{27 + \frac{12 \sqrt{3165}}{25}}}\right)$$
=
-2
$$-2$$
-2
Respuesta numérica
[src]
x1 = 0.603520046963645 + 1.13697376454384*i
x2 = -1.20704009392729
x3 = 0.603520046963645 - 1.13697376454384*i
x3 = 0.603520046963645 - 1.13697376454384*i