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Ecuación x^2-35=2*x
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Ecuación (x+2)(-x+6)=0
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Ecuación (x+2)+x=9
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Ecuación 2*x^3+6*x+9=0
- Expresar {x} en función de y en la ecuación:
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Expresiones idénticas
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- 2*x en el grado 3+6*x+9=0
- 2x^3+6x+9=0
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- 2*x^3+6*x+9=O
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Expresiones semejantes
- 2*x^3-6*x+9=0
- 2*x^3+6*x-9=0
2*x^3+6*x+9=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(2 x^{3} + 6 x\right) + 9 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + 3 x + \frac{9}{2} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 3$$
$$v = \frac{d}{a}$$
$$v = \frac{9}{2}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 3$$
$$x_{1} x_{2} x_{3} = \frac{9}{2}$$
$$\left(2 x^{3} + 6 x\right) + 9 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} + 3 x + \frac{9}{2} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 3$$
$$v = \frac{d}{a}$$
$$v = \frac{9}{2}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 3$$
$$x_{1} x_{2} x_{3} = \frac{9}{2}$$
Suma y producto de raíces
[src]
suma
_________________ / _________________ \ _________________ / _________________\ _________________
/ ____ | / ____ | / ____ | / ____ | / ____
/ 243 27*\/ 97 | ___ / 243 27*\/ 97 | / 243 27*\/ 97 | ___ / 243 27*\/ 97 | / 243 27*\/ 97
3 / --- + --------- |\/ 3 *3 / --- + --------- ___ | 3 / --- + --------- | ___ \/ 3 *3 / --- + --------- | 3 / --- + ---------
3 \/ 4 4 | \/ 4 4 3*\/ 3 | 3 \/ 4 4 | 3*\/ 3 \/ 4 4 | 3 \/ 4 4
- ------------------------ + ---------------------- + I*|---------------------------- + ------------------------| + - ------------------------ + ---------------------- + I*|- ------------------------ - ----------------------------| + ---------------------- - ----------------------
_________________ 6 | 6 _________________| _________________ 6 | _________________ 6 | _________________ 3
/ ____ | / ____ | / ____ | / ____ | / ____
/ 243 27*\/ 97 | / 243 27*\/ 97 | / 243 27*\/ 97 | / 243 27*\/ 97 | / 243 27*\/ 97
2*3 / --- + --------- | 2*3 / --- + --------- | 2*3 / --- + --------- | 2*3 / --- + --------- | 3 / --- + ---------
\/ 4 4 \ \/ 4 4 / \/ 4 4 \ \/ 4 4 / \/ 4 4
$$\left(- \frac{\sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}}{3} + \frac{3}{\sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}}\right) + \left(\left(- \frac{3}{2 \sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}} + \frac{\sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}}{6} - \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}}\right)\right) + \left(- \frac{3}{2 \sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}} + \frac{\sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}}{6} + i \left(\frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}} + \frac{\sqrt{3} \sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}}{6}\right)\right)\right)$$
=
/ _________________\ / _________________ \ | / ____ | | / ____ | | ___ / 243 27*\/ 97 | | ___ / 243 27*\/ 97 | | ___ \/ 3 *3 / --- + --------- | |\/ 3 *3 / --- + --------- ___ | | 3*\/ 3 \/ 4 4 | | \/ 4 4 3*\/ 3 | I*|- ------------------------ - ----------------------------| + I*|---------------------------- + ------------------------| | _________________ 6 | | 6 _________________| | / ____ | | / ____ | | / 243 27*\/ 97 | | / 243 27*\/ 97 | | 2*3 / --- + --------- | | 2*3 / --- + --------- | \ \/ 4 4 / \ \/ 4 4 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}}{6} - \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}}\right) + i \left(\frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}} + \frac{\sqrt{3} \sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}}{6}\right)$$
producto
/ _________________ / _________________ \\ / _________________ / _________________\\ / _________________\ | / ____ | / ____ || | / ____ | / ____ || | / ____ | | / 243 27*\/ 97 | ___ / 243 27*\/ 97 || | / 243 27*\/ 97 | ___ / 243 27*\/ 97 || | / 243 27*\/ 97 | | 3 / --- + --------- |\/ 3 *3 / --- + --------- ___ || | 3 / --- + --------- | ___ \/ 3 *3 / --- + --------- || | 3 / --- + --------- | | 3 \/ 4 4 | \/ 4 4 3*\/ 3 || | 3 \/ 4 4 | 3*\/ 3 \/ 4 4 || | 3 \/ 4 4 | |- ------------------------ + ---------------------- + I*|---------------------------- + ------------------------||*|- ------------------------ + ---------------------- + I*|- ------------------------ - ----------------------------||*|---------------------- - ----------------------| | _________________ 6 | 6 _________________|| | _________________ 6 | _________________ 6 || | _________________ 3 | | / ____ | / ____ || | / ____ | / ____ || | / ____ | | / 243 27*\/ 97 | / 243 27*\/ 97 || | / 243 27*\/ 97 | / 243 27*\/ 97 || | / 243 27*\/ 97 | | 2*3 / --- + --------- | 2*3 / --- + --------- || | 2*3 / --- + --------- | 2*3 / --- + --------- || |3 / --- + --------- | \ \/ 4 4 \ \/ 4 4 // \ \/ 4 4 \ \/ 4 4 // \\/ 4 4 /
$$\left(- \frac{3}{2 \sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}} + \frac{\sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}}{6} + i \left(\frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}} + \frac{\sqrt{3} \sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}}{6}\right)\right) \left(- \frac{3}{2 \sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}} + \frac{\sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}}{6} - \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}}\right)\right) \left(- \frac{\sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}}{3} + \frac{3}{\sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}}\right)$$
=
-9/2
$$- \frac{9}{2}$$
-9/2
Respuesta rápida
[src]
_________________ / _________________ \
/ ____ | / ____ |
/ 243 27*\/ 97 | ___ / 243 27*\/ 97 |
3 / --- + --------- |\/ 3 *3 / --- + --------- ___ |
3 \/ 4 4 | \/ 4 4 3*\/ 3 |
x1 = - ------------------------ + ---------------------- + I*|---------------------------- + ------------------------|
_________________ 6 | 6 _________________|
/ ____ | / ____ |
/ 243 27*\/ 97 | / 243 27*\/ 97 |
2*3 / --- + --------- | 2*3 / --- + --------- |
\/ 4 4 \ \/ 4 4 /
$$x_{1} = - \frac{3}{2 \sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}} + \frac{\sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}}{6} + i \left(\frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}} + \frac{\sqrt{3} \sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}}{6}\right)$$
_________________ / _________________\
/ ____ | / ____ |
/ 243 27*\/ 97 | ___ / 243 27*\/ 97 |
3 / --- + --------- | ___ \/ 3 *3 / --- + --------- |
3 \/ 4 4 | 3*\/ 3 \/ 4 4 |
x2 = - ------------------------ + ---------------------- + I*|- ------------------------ - ----------------------------|
_________________ 6 | _________________ 6 |
/ ____ | / ____ |
/ 243 27*\/ 97 | / 243 27*\/ 97 |
2*3 / --- + --------- | 2*3 / --- + --------- |
\/ 4 4 \ \/ 4 4 /
$$x_{2} = - \frac{3}{2 \sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}} + \frac{\sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}}{6} - \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}}\right)$$
_________________
/ ____
/ 243 27*\/ 97
3 / --- + ---------
3 \/ 4 4
x3 = ---------------------- - ----------------------
_________________ 3
/ ____
/ 243 27*\/ 97
3 / --- + ---------
\/ 4 4
$$x_{3} = - \frac{\sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}}{3} + \frac{3}{\sqrt[3]{\frac{243}{4} + \frac{27 \sqrt{97}}{4}}}$$
x3 = -(243/4 + 27*sqrt(97)/4)^(1/3)/3 + 3/(243/4 + 27*sqrt(97)/4)^(1/3)
Respuesta numérica
[src]
x1 = 0.540022156083668 - 1.96846940214568*i
x2 = -1.08004431216734
x3 = 0.540022156083668 + 1.96846940214568*i
x3 = 0.540022156083668 + 1.96846940214568*i
Gráfico