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Expresiones semejantes
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x+x^3-2*x^2+3=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
3 2 x + x - 2*x + 3 = 0
$$\left(- 2 x^{2} + \left(x^{3} + x\right)\right) + 3 = 0$$
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -2$$
$$q = \frac{c}{a}$$
$$q = 1$$
$$v = \frac{d}{a}$$
$$v = 3$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 2$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 1$$
$$x_{1} x_{2} x_{3} = 3$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -2$$
$$q = \frac{c}{a}$$
$$q = 1$$
$$v = \frac{d}{a}$$
$$v = 3$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 2$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 1$$
$$x_{1} x_{2} x_{3} = 3$$
Suma y producto de raíces
[src]
suma
_______________ / _______________\ _______________ / _______________ \ _______________
/ ____ | / ____ | / ____ | / ____ | / ____
/ 83 9*\/ 85 | ___ / 83 9*\/ 85 | / 83 9*\/ 85 | ___ / 83 9*\/ 85 | / 83 9*\/ 85
3 / -- + -------- | ___ \/ 3 *3 / -- + -------- | 3 / -- + -------- | \/ 3 *3 / -- + -------- ___ | 3 / -- + --------
2 1 \/ 2 2 | \/ 3 \/ 2 2 | 2 1 \/ 2 2 | \/ 2 2 \/ 3 | 2 1 \/ 2 2
- + ---------------------- + -------------------- + I*|- ---------------------- + --------------------------| + - + ---------------------- + -------------------- + I*|- -------------------------- + ----------------------| + - - ---------------------- - --------------------
3 _______________ 6 | _______________ 6 | 3 _______________ 6 | 6 _______________| 3 _______________ 3
/ ____ | / ____ | / ____ | / ____ | / ____
/ 83 9*\/ 85 | / 83 9*\/ 85 | / 83 9*\/ 85 | / 83 9*\/ 85 | / 83 9*\/ 85
6*3 / -- + -------- | 6*3 / -- + -------- | 6*3 / -- + -------- | 6*3 / -- + -------- | 3*3 / -- + --------
\/ 2 2 \ \/ 2 2 / \/ 2 2 \ \/ 2 2 / \/ 2 2
$$\left(- \frac{\sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}}{3} - \frac{1}{3 \sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}} + \frac{2}{3}\right) + \left(\left(\frac{1}{6 \sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}} + \frac{2}{3} + \frac{\sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}}{6} + \frac{\sqrt{3}}{6 \sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}}\right)\right) + \left(\frac{1}{6 \sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}} + \frac{2}{3} + \frac{\sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}}{6} + i \left(- \frac{\sqrt{3}}{6 \sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}}{6}\right)\right)\right)$$
=
/ _______________\ / _______________ \
| / ____ | | / ____ |
| ___ / 83 9*\/ 85 | | ___ / 83 9*\/ 85 |
| ___ \/ 3 *3 / -- + -------- | | \/ 3 *3 / -- + -------- ___ |
| \/ 3 \/ 2 2 | | \/ 2 2 \/ 3 |
2 + I*|- ---------------------- + --------------------------| + I*|- -------------------------- + ----------------------|
| _______________ 6 | | 6 _______________|
| / ____ | | / ____ |
| / 83 9*\/ 85 | | / 83 9*\/ 85 |
| 6*3 / -- + -------- | | 6*3 / -- + -------- |
\ \/ 2 2 / \ \/ 2 2 /
$$2 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}}{6} + \frac{\sqrt{3}}{6 \sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}}\right) + i \left(- \frac{\sqrt{3}}{6 \sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}}{6}\right)$$
producto
/ _______________ / _______________\\ / _______________ / _______________ \\ / _______________\ | / ____ | / ____ || | / ____ | / ____ || | / ____ | | / 83 9*\/ 85 | ___ / 83 9*\/ 85 || | / 83 9*\/ 85 | ___ / 83 9*\/ 85 || | / 83 9*\/ 85 | | 3 / -- + -------- | ___ \/ 3 *3 / -- + -------- || | 3 / -- + -------- | \/ 3 *3 / -- + -------- ___ || | 3 / -- + -------- | |2 1 \/ 2 2 | \/ 3 \/ 2 2 || |2 1 \/ 2 2 | \/ 2 2 \/ 3 || |2 1 \/ 2 2 | |- + ---------------------- + -------------------- + I*|- ---------------------- + --------------------------||*|- + ---------------------- + -------------------- + I*|- -------------------------- + ----------------------||*|- - ---------------------- - --------------------| |3 _______________ 6 | _______________ 6 || |3 _______________ 6 | 6 _______________|| |3 _______________ 3 | | / ____ | / ____ || | / ____ | / ____ || | / ____ | | / 83 9*\/ 85 | / 83 9*\/ 85 || | / 83 9*\/ 85 | / 83 9*\/ 85 || | / 83 9*\/ 85 | | 6*3 / -- + -------- | 6*3 / -- + -------- || | 6*3 / -- + -------- | 6*3 / -- + -------- || | 3*3 / -- + -------- | \ \/ 2 2 \ \/ 2 2 // \ \/ 2 2 \ \/ 2 2 // \ \/ 2 2 /
$$\left(\frac{1}{6 \sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}} + \frac{2}{3} + \frac{\sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}}{6} + i \left(- \frac{\sqrt{3}}{6 \sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}}{6}\right)\right) \left(\frac{1}{6 \sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}} + \frac{2}{3} + \frac{\sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}}{6} + \frac{\sqrt{3}}{6 \sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}}\right)\right) \left(- \frac{\sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}}{3} - \frac{1}{3 \sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}} + \frac{2}{3}\right)$$
=
-3
$$-3$$
-3
Respuesta rápida
[src]
_______________ / _______________\
/ ____ | / ____ |
/ 83 9*\/ 85 | ___ / 83 9*\/ 85 |
3 / -- + -------- | ___ \/ 3 *3 / -- + -------- |
2 1 \/ 2 2 | \/ 3 \/ 2 2 |
x1 = - + ---------------------- + -------------------- + I*|- ---------------------- + --------------------------|
3 _______________ 6 | _______________ 6 |
/ ____ | / ____ |
/ 83 9*\/ 85 | / 83 9*\/ 85 |
6*3 / -- + -------- | 6*3 / -- + -------- |
\/ 2 2 \ \/ 2 2 /
$$x_{1} = \frac{1}{6 \sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}} + \frac{2}{3} + \frac{\sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}}{6} + i \left(- \frac{\sqrt{3}}{6 \sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}}{6}\right)$$
_______________ / _______________ \
/ ____ | / ____ |
/ 83 9*\/ 85 | ___ / 83 9*\/ 85 |
3 / -- + -------- | \/ 3 *3 / -- + -------- ___ |
2 1 \/ 2 2 | \/ 2 2 \/ 3 |
x2 = - + ---------------------- + -------------------- + I*|- -------------------------- + ----------------------|
3 _______________ 6 | 6 _______________|
/ ____ | / ____ |
/ 83 9*\/ 85 | / 83 9*\/ 85 |
6*3 / -- + -------- | 6*3 / -- + -------- |
\/ 2 2 \ \/ 2 2 /
$$x_{2} = \frac{1}{6 \sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}} + \frac{2}{3} + \frac{\sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}}{6} + \frac{\sqrt{3}}{6 \sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}}\right)$$
_______________
/ ____
/ 83 9*\/ 85
3 / -- + --------
2 1 \/ 2 2
x3 = - - ---------------------- - --------------------
3 _______________ 3
/ ____
/ 83 9*\/ 85
3*3 / -- + --------
\/ 2 2
$$x_{3} = - \frac{\sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}}{3} - \frac{1}{3 \sqrt[3]{\frac{9 \sqrt{85}}{2} + \frac{83}{2}}} + \frac{2}{3}$$
x3 = -(9*sqrt(85)/2 + 83/2)^(1/3)/3 - 1/(3*(9*sqrt(85)/2 + 83/2)^(1/3)) + 2/3
Respuesta numérica
[src]
x1 = 1.43185326390959 + 1.19297873093668*i
x2 = -0.863706527819189
x3 = 1.43185326390959 - 1.19297873093668*i
x3 = 1.43185326390959 - 1.19297873093668*i
Gráfico