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xxx-2.5x+2.5=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
5*x 5
x*x*x - --- + - = 0
2 2
$$\left(x x x - \frac{5 x}{2}\right) + \frac{5}{2} = 0$$
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = - \frac{5}{2}$$
$$v = \frac{d}{a}$$
$$v = \frac{5}{2}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = - \frac{5}{2}$$
$$x_{1} x_{2} x_{3} = \frac{5}{2}$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = - \frac{5}{2}$$
$$v = \frac{d}{a}$$
$$v = \frac{5}{2}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = - \frac{5}{2}$$
$$x_{1} x_{2} x_{3} = \frac{5}{2}$$
Respuesta rápida
[src]
_________________ / _________________\
/ ____ | / ____ |
/ 135 15*\/ 51 | ___ / 135 15*\/ 51 |
3 / --- + --------- | ___ \/ 3 *3 / --- + --------- |
\/ 4 4 5 | 5*\/ 3 \/ 4 4 |
x1 = ---------------------- + ------------------------ + I*|- ------------------------ + ----------------------------|
6 _________________ | _________________ 6 |
/ ____ | / ____ |
/ 135 15*\/ 51 | / 135 15*\/ 51 |
4*3 / --- + --------- | 4*3 / --- + --------- |
\/ 4 4 \ \/ 4 4 /
$$x_{1} = \frac{5}{4 \sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}} + \frac{\sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}}{6} + i \left(- \frac{5 \sqrt{3}}{4 \sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}} + \frac{\sqrt{3} \sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}}{6}\right)$$
_________________ / _________________ \
/ ____ | / ____ |
/ 135 15*\/ 51 | ___ / 135 15*\/ 51 |
3 / --- + --------- | \/ 3 *3 / --- + --------- ___ |
\/ 4 4 5 | \/ 4 4 5*\/ 3 |
x2 = ---------------------- + ------------------------ + I*|- ---------------------------- + ------------------------|
6 _________________ | 6 _________________|
/ ____ | / ____ |
/ 135 15*\/ 51 | / 135 15*\/ 51 |
4*3 / --- + --------- | 4*3 / --- + --------- |
\/ 4 4 \ \/ 4 4 /
$$x_{2} = \frac{5}{4 \sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}} + \frac{\sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}}{6} + \frac{5 \sqrt{3}}{4 \sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}}\right)$$
_________________
/ ____
/ 135 15*\/ 51
3 / --- + ---------
5 \/ 4 4
x3 = - ------------------------ - ----------------------
_________________ 3
/ ____
/ 135 15*\/ 51
2*3 / --- + ---------
\/ 4 4
$$x_{3} = - \frac{\sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}}{3} - \frac{5}{2 \sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}}$$
x3 = -(15*sqrt(51)/4 + 135/4)^(1/3)/3 - 5/(2*(15*sqrt(51)/4 + 135/4)^(1/3))
Suma y producto de raíces
[src]
suma
_________________ / _________________\ _________________ / _________________ \ _________________
/ ____ | / ____ | / ____ | / ____ | / ____
/ 135 15*\/ 51 | ___ / 135 15*\/ 51 | / 135 15*\/ 51 | ___ / 135 15*\/ 51 | / 135 15*\/ 51
3 / --- + --------- | ___ \/ 3 *3 / --- + --------- | 3 / --- + --------- | \/ 3 *3 / --- + --------- ___ | 3 / --- + ---------
\/ 4 4 5 | 5*\/ 3 \/ 4 4 | \/ 4 4 5 | \/ 4 4 5*\/ 3 | 5 \/ 4 4
---------------------- + ------------------------ + I*|- ------------------------ + ----------------------------| + ---------------------- + ------------------------ + I*|- ---------------------------- + ------------------------| + - ------------------------ - ----------------------
6 _________________ | _________________ 6 | 6 _________________ | 6 _________________| _________________ 3
/ ____ | / ____ | / ____ | / ____ | / ____
/ 135 15*\/ 51 | / 135 15*\/ 51 | / 135 15*\/ 51 | / 135 15*\/ 51 | / 135 15*\/ 51
4*3 / --- + --------- | 4*3 / --- + --------- | 4*3 / --- + --------- | 4*3 / --- + --------- | 2*3 / --- + ---------
\/ 4 4 \ \/ 4 4 / \/ 4 4 \ \/ 4 4 / \/ 4 4
$$\left(- \frac{\sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}}{3} - \frac{5}{2 \sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}}\right) + \left(\left(\frac{5}{4 \sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}} + \frac{\sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}}{6} + \frac{5 \sqrt{3}}{4 \sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}}\right)\right) + \left(\frac{5}{4 \sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}} + \frac{\sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}}{6} + i \left(- \frac{5 \sqrt{3}}{4 \sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}} + \frac{\sqrt{3} \sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}}{6}\right)\right)\right)$$
=
/ _________________\ / _________________ \ | / ____ | | / ____ | | ___ / 135 15*\/ 51 | | ___ / 135 15*\/ 51 | | ___ \/ 3 *3 / --- + --------- | | \/ 3 *3 / --- + --------- ___ | | 5*\/ 3 \/ 4 4 | | \/ 4 4 5*\/ 3 | I*|- ------------------------ + ----------------------------| + I*|- ---------------------------- + ------------------------| | _________________ 6 | | 6 _________________| | / ____ | | / ____ | | / 135 15*\/ 51 | | / 135 15*\/ 51 | | 4*3 / --- + --------- | | 4*3 / --- + --------- | \ \/ 4 4 / \ \/ 4 4 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}}{6} + \frac{5 \sqrt{3}}{4 \sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}}\right) + i \left(- \frac{5 \sqrt{3}}{4 \sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}} + \frac{\sqrt{3} \sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}}{6}\right)$$
producto
/ _________________ / _________________\\ / _________________ / _________________ \\ / _________________\ | / ____ | / ____ || | / ____ | / ____ || | / ____ | | / 135 15*\/ 51 | ___ / 135 15*\/ 51 || | / 135 15*\/ 51 | ___ / 135 15*\/ 51 || | / 135 15*\/ 51 | |3 / --- + --------- | ___ \/ 3 *3 / --- + --------- || |3 / --- + --------- | \/ 3 *3 / --- + --------- ___ || | 3 / --- + --------- | |\/ 4 4 5 | 5*\/ 3 \/ 4 4 || |\/ 4 4 5 | \/ 4 4 5*\/ 3 || | 5 \/ 4 4 | |---------------------- + ------------------------ + I*|- ------------------------ + ----------------------------||*|---------------------- + ------------------------ + I*|- ---------------------------- + ------------------------||*|- ------------------------ - ----------------------| | 6 _________________ | _________________ 6 || | 6 _________________ | 6 _________________|| | _________________ 3 | | / ____ | / ____ || | / ____ | / ____ || | / ____ | | / 135 15*\/ 51 | / 135 15*\/ 51 || | / 135 15*\/ 51 | / 135 15*\/ 51 || | / 135 15*\/ 51 | | 4*3 / --- + --------- | 4*3 / --- + --------- || | 4*3 / --- + --------- | 4*3 / --- + --------- || | 2*3 / --- + --------- | \ \/ 4 4 \ \/ 4 4 // \ \/ 4 4 \ \/ 4 4 // \ \/ 4 4 /
$$\left(\frac{5}{4 \sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}} + \frac{\sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}}{6} + i \left(- \frac{5 \sqrt{3}}{4 \sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}} + \frac{\sqrt{3} \sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}}{6}\right)\right) \left(\frac{5}{4 \sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}} + \frac{\sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}}{6} + \frac{5 \sqrt{3}}{4 \sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}}\right)\right) \left(- \frac{\sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}}{3} - \frac{5}{2 \sqrt[3]{\frac{15 \sqrt{51}}{4} + \frac{135}{4}}}\right)$$
=
-5/2
$$- \frac{5}{2}$$
-5/2
Respuesta numérica
[src]
x1 = -1.94551020654183
x2 = 0.972755103270916 - 0.582028756006808*i
x3 = 0.972755103270916 + 0.582028756006808*i
x3 = 0.972755103270916 + 0.582028756006808*i