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- 9x+x^3+6x^2+8=0
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Expresiones semejantes
- 9*x-x^3+6*x^2+8=0
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9*x+x^3+6*x^2+8=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
3 2 9*x + x + 6*x + 8 = 0
$$\left(6 x^{2} + \left(x^{3} + 9 x\right)\right) + 8 = 0$$
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 6$$
$$q = \frac{c}{a}$$
$$q = 9$$
$$v = \frac{d}{a}$$
$$v = 8$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -6$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 9$$
$$x_{1} x_{2} x_{3} = 8$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 6$$
$$q = \frac{c}{a}$$
$$q = 9$$
$$v = \frac{d}{a}$$
$$v = 8$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -6$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 9$$
$$x_{1} x_{2} x_{3} = 8$$
Suma y producto de raíces
[src]
suma
_______________ / _______________\ _______________ / _______________ \ _______________
3 / ___ | ___ ___ 3 / ___ | 3 / ___ | ___ 3 / ___ ___ | 3 / ___
\/ 81 + 54*\/ 2 3 | 3*\/ 3 \/ 3 *\/ 81 + 54*\/ 2 | \/ 81 + 54*\/ 2 3 | \/ 3 *\/ 81 + 54*\/ 2 3*\/ 3 | 3 \/ 81 + 54*\/ 2
-2 + ------------------ + -------------------- + I*|- -------------------- + ------------------------| + -2 + ------------------ + -------------------- + I*|- ------------------------ + --------------------| + -2 - ------------------ - ------------------
6 _______________ | _______________ 6 | 6 _______________ | 6 _______________| _______________ 3
3 / ___ | 3 / ___ | 3 / ___ | 3 / ___ | 3 / ___
2*\/ 81 + 54*\/ 2 \ 2*\/ 81 + 54*\/ 2 / 2*\/ 81 + 54*\/ 2 \ 2*\/ 81 + 54*\/ 2 / \/ 81 + 54*\/ 2
$$\left(-2 - \frac{\sqrt[3]{54 \sqrt{2} + 81}}{3} - \frac{3}{\sqrt[3]{54 \sqrt{2} + 81}}\right) + \left(\left(-2 + \frac{3}{2 \sqrt[3]{54 \sqrt{2} + 81}} + \frac{\sqrt[3]{54 \sqrt{2} + 81}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{54 \sqrt{2} + 81}}{6} + \frac{3 \sqrt{3}}{2 \sqrt[3]{54 \sqrt{2} + 81}}\right)\right) + \left(-2 + \frac{3}{2 \sqrt[3]{54 \sqrt{2} + 81}} + \frac{\sqrt[3]{54 \sqrt{2} + 81}}{6} + i \left(- \frac{3 \sqrt{3}}{2 \sqrt[3]{54 \sqrt{2} + 81}} + \frac{\sqrt{3} \sqrt[3]{54 \sqrt{2} + 81}}{6}\right)\right)\right)$$
=
/ _______________\ / _______________ \
| ___ ___ 3 / ___ | | ___ 3 / ___ ___ |
| 3*\/ 3 \/ 3 *\/ 81 + 54*\/ 2 | | \/ 3 *\/ 81 + 54*\/ 2 3*\/ 3 |
-6 + I*|- -------------------- + ------------------------| + I*|- ------------------------ + --------------------|
| _______________ 6 | | 6 _______________|
| 3 / ___ | | 3 / ___ |
\ 2*\/ 81 + 54*\/ 2 / \ 2*\/ 81 + 54*\/ 2 /
$$-6 + i \left(- \frac{\sqrt{3} \sqrt[3]{54 \sqrt{2} + 81}}{6} + \frac{3 \sqrt{3}}{2 \sqrt[3]{54 \sqrt{2} + 81}}\right) + i \left(- \frac{3 \sqrt{3}}{2 \sqrt[3]{54 \sqrt{2} + 81}} + \frac{\sqrt{3} \sqrt[3]{54 \sqrt{2} + 81}}{6}\right)$$
producto
/ _______________ / _______________\\ / _______________ / _______________ \\ / _______________\ | 3 / ___ | ___ ___ 3 / ___ || | 3 / ___ | ___ 3 / ___ ___ || | 3 / ___ | | \/ 81 + 54*\/ 2 3 | 3*\/ 3 \/ 3 *\/ 81 + 54*\/ 2 || | \/ 81 + 54*\/ 2 3 | \/ 3 *\/ 81 + 54*\/ 2 3*\/ 3 || | 3 \/ 81 + 54*\/ 2 | |-2 + ------------------ + -------------------- + I*|- -------------------- + ------------------------||*|-2 + ------------------ + -------------------- + I*|- ------------------------ + --------------------||*|-2 - ------------------ - ------------------| | 6 _______________ | _______________ 6 || | 6 _______________ | 6 _______________|| | _______________ 3 | | 3 / ___ | 3 / ___ || | 3 / ___ | 3 / ___ || | 3 / ___ | \ 2*\/ 81 + 54*\/ 2 \ 2*\/ 81 + 54*\/ 2 // \ 2*\/ 81 + 54*\/ 2 \ 2*\/ 81 + 54*\/ 2 // \ \/ 81 + 54*\/ 2 /
$$\left(-2 + \frac{3}{2 \sqrt[3]{54 \sqrt{2} + 81}} + \frac{\sqrt[3]{54 \sqrt{2} + 81}}{6} + i \left(- \frac{3 \sqrt{3}}{2 \sqrt[3]{54 \sqrt{2} + 81}} + \frac{\sqrt{3} \sqrt[3]{54 \sqrt{2} + 81}}{6}\right)\right) \left(-2 + \frac{3}{2 \sqrt[3]{54 \sqrt{2} + 81}} + \frac{\sqrt[3]{54 \sqrt{2} + 81}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{54 \sqrt{2} + 81}}{6} + \frac{3 \sqrt{3}}{2 \sqrt[3]{54 \sqrt{2} + 81}}\right)\right) \left(-2 - \frac{\sqrt[3]{54 \sqrt{2} + 81}}{3} - \frac{3}{\sqrt[3]{54 \sqrt{2} + 81}}\right)$$
=
-8
$$-8$$
-8
Respuesta rápida
[src]
_______________ / _______________\
3 / ___ | ___ ___ 3 / ___ |
\/ 81 + 54*\/ 2 3 | 3*\/ 3 \/ 3 *\/ 81 + 54*\/ 2 |
x1 = -2 + ------------------ + -------------------- + I*|- -------------------- + ------------------------|
6 _______________ | _______________ 6 |
3 / ___ | 3 / ___ |
2*\/ 81 + 54*\/ 2 \ 2*\/ 81 + 54*\/ 2 /
$$x_{1} = -2 + \frac{3}{2 \sqrt[3]{54 \sqrt{2} + 81}} + \frac{\sqrt[3]{54 \sqrt{2} + 81}}{6} + i \left(- \frac{3 \sqrt{3}}{2 \sqrt[3]{54 \sqrt{2} + 81}} + \frac{\sqrt{3} \sqrt[3]{54 \sqrt{2} + 81}}{6}\right)$$
_______________ / _______________ \
3 / ___ | ___ 3 / ___ ___ |
\/ 81 + 54*\/ 2 3 | \/ 3 *\/ 81 + 54*\/ 2 3*\/ 3 |
x2 = -2 + ------------------ + -------------------- + I*|- ------------------------ + --------------------|
6 _______________ | 6 _______________|
3 / ___ | 3 / ___ |
2*\/ 81 + 54*\/ 2 \ 2*\/ 81 + 54*\/ 2 /
$$x_{2} = -2 + \frac{3}{2 \sqrt[3]{54 \sqrt{2} + 81}} + \frac{\sqrt[3]{54 \sqrt{2} + 81}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{54 \sqrt{2} + 81}}{6} + \frac{3 \sqrt{3}}{2 \sqrt[3]{54 \sqrt{2} + 81}}\right)$$
_______________
3 / ___
3 \/ 81 + 54*\/ 2
x3 = -2 - ------------------ - ------------------
_______________ 3
3 / ___
\/ 81 + 54*\/ 2
$$x_{3} = -2 - \frac{\sqrt[3]{54 \sqrt{2} + 81}}{3} - \frac{3}{\sqrt[3]{54 \sqrt{2} + 81}}$$
x3 = -2 - (54*sqrt(2) + 81)^(1/3)/3 - 3/(54*sqrt(2) + 81)^(1/3)
Respuesta numérica
[src]
x1 = -4.35530139760812
x2 = -0.82234930119594 - 1.07730381284996*i
x3 = -0.82234930119594 + 1.07730381284996*i
x3 = -0.82234930119594 + 1.07730381284996*i
Gráfico