36*x^3-42*x^2-60*x+11=0 la ecuación
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Solución
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(- 60 x + \left(36 x^{3} - 42 x^{2}\right)\right) + 11 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - \frac{7 x^{2}}{6} - \frac{5 x}{3} + \frac{11}{36} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - \frac{7}{6}$$
$$q = \frac{c}{a}$$
$$q = - \frac{5}{3}$$
$$v = \frac{d}{a}$$
$$v = \frac{11}{36}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = \frac{7}{6}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = - \frac{5}{3}$$
$$x_{1} x_{2} x_{3} = \frac{11}{36}$$
$$x_{1} = \frac{1}{6}$$
___
1 5*\/ 3
x2 = - - -------
2 6
$$x_{2} = \frac{1}{2} - \frac{5 \sqrt{3}}{6}$$
___
1 5*\/ 3
x3 = - + -------
2 6
$$x_{3} = \frac{1}{2} + \frac{5 \sqrt{3}}{6}$$
Suma y producto de raíces
[src]
___ ___
1 1 5*\/ 3 1 5*\/ 3
- + - - ------- + - + -------
6 2 6 2 6
$$\left(\left(\frac{1}{2} - \frac{5 \sqrt{3}}{6}\right) + \frac{1}{6}\right) + \left(\frac{1}{2} + \frac{5 \sqrt{3}}{6}\right)$$
$$\frac{7}{6}$$
___
1 5*\/ 3
- - ------- / ___\
2 6 |1 5*\/ 3 |
-----------*|- + -------|
6 \2 6 /
$$\frac{\frac{1}{2} - \frac{5 \sqrt{3}}{6}}{6} \left(\frac{1}{2} + \frac{5 \sqrt{3}}{6}\right)$$
$$- \frac{11}{36}$$