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- La ecuación:
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Ecuación x^3-x^2-x-1=0
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Ecuación 12-4(x-3)=39-9x
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Ecuación 0,2x+2,7=1,4-1,1x
- Expresar {x} en función de y en la ecuación:
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- 2*x-15*y=-1
- -18*x-6*y=-1
- 6*x+9*y=-9
- Gráfico de la función y =:
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x^3+2*x^2+1
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Expresiones idénticas
- x^ tres + dos *x^ dos + uno = cero
- x al cubo más 2 multiplicar por x al cuadrado más 1 es igual a 0
- x en el grado tres más dos multiplicar por x en el grado dos más uno es igual a cero
- x3+2*x2+1=0
- x³+2*x²+1=0
- x en el grado 3+2*x en el grado 2+1=0
- x^3+2x^2+1=0
- x3+2x2+1=0
- x^3+2*x^2+1=O
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Expresiones semejantes
- x^3-2*x^2+1=0
- x^3+2*x^2-1=0
x^3+2*x^2+1=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 2$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = 1$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -2$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = 1$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 2$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = 1$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -2$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = 1$$
Suma y producto de raíces
[src]
suma
________________ / ________________\ ________________ / ________________ \ ________________
/ _____ | / _____ | / _____ | / _____ | / _____
/ 43 3*\/ 177 | ___ / 43 3*\/ 177 | / 43 3*\/ 177 | ___ / 43 3*\/ 177 | / 43 3*\/ 177
3 / -- + --------- | ___ \/ 3 *3 / -- + --------- | 3 / -- + --------- | \/ 3 *3 / -- + --------- ___ | 3 / -- + ---------
2 \/ 2 2 2 | 2*\/ 3 \/ 2 2 | 2 \/ 2 2 2 | \/ 2 2 2*\/ 3 | 2 4 \/ 2 2
- - + --------------------- + ----------------------- + I*|- ----------------------- + ---------------------------| + - - + --------------------- + ----------------------- + I*|- --------------------------- + -----------------------| + - - - ----------------------- - ---------------------
3 6 ________________ | ________________ 6 | 3 6 ________________ | 6 ________________| 3 ________________ 3
/ _____ | / _____ | / _____ | / _____ | / _____
/ 43 3*\/ 177 | / 43 3*\/ 177 | / 43 3*\/ 177 | / 43 3*\/ 177 | / 43 3*\/ 177
3*3 / -- + --------- | 3*3 / -- + --------- | 3*3 / -- + --------- | 3*3 / -- + --------- | 3*3 / -- + ---------
\/ 2 2 \ \/ 2 2 / \/ 2 2 \ \/ 2 2 / \/ 2 2
$$\left(- \frac{\sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{3} - \frac{2}{3} - \frac{4}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}\right) + \left(\left(- \frac{2}{3} + \frac{2}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{6} + \frac{2 \sqrt{3}}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}\right)\right) + \left(- \frac{2}{3} + \frac{2}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{6} + i \left(- \frac{2 \sqrt{3}}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{6}\right)\right)\right)$$
=
/ ________________\ / ________________ \
| / _____ | | / _____ |
| ___ / 43 3*\/ 177 | | ___ / 43 3*\/ 177 |
| ___ \/ 3 *3 / -- + --------- | | \/ 3 *3 / -- + --------- ___ |
| 2*\/ 3 \/ 2 2 | | \/ 2 2 2*\/ 3 |
-2 + I*|- ----------------------- + ---------------------------| + I*|- --------------------------- + -----------------------|
| ________________ 6 | | 6 ________________|
| / _____ | | / _____ |
| / 43 3*\/ 177 | | / 43 3*\/ 177 |
| 3*3 / -- + --------- | | 3*3 / -- + --------- |
\ \/ 2 2 / \ \/ 2 2 /
$$-2 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{6} + \frac{2 \sqrt{3}}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}\right) + i \left(- \frac{2 \sqrt{3}}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{6}\right)$$
producto
/ ________________ / ________________\\ / ________________ / ________________ \\ / ________________\ | / _____ | / _____ || | / _____ | / _____ || | / _____ | | / 43 3*\/ 177 | ___ / 43 3*\/ 177 || | / 43 3*\/ 177 | ___ / 43 3*\/ 177 || | / 43 3*\/ 177 | | 3 / -- + --------- | ___ \/ 3 *3 / -- + --------- || | 3 / -- + --------- | \/ 3 *3 / -- + --------- ___ || | 3 / -- + --------- | | 2 \/ 2 2 2 | 2*\/ 3 \/ 2 2 || | 2 \/ 2 2 2 | \/ 2 2 2*\/ 3 || | 2 4 \/ 2 2 | |- - + --------------------- + ----------------------- + I*|- ----------------------- + ---------------------------||*|- - + --------------------- + ----------------------- + I*|- --------------------------- + -----------------------||*|- - - ----------------------- - ---------------------| | 3 6 ________________ | ________________ 6 || | 3 6 ________________ | 6 ________________|| | 3 ________________ 3 | | / _____ | / _____ || | / _____ | / _____ || | / _____ | | / 43 3*\/ 177 | / 43 3*\/ 177 || | / 43 3*\/ 177 | / 43 3*\/ 177 || | / 43 3*\/ 177 | | 3*3 / -- + --------- | 3*3 / -- + --------- || | 3*3 / -- + --------- | 3*3 / -- + --------- || | 3*3 / -- + --------- | \ \/ 2 2 \ \/ 2 2 // \ \/ 2 2 \ \/ 2 2 // \ \/ 2 2 /
$$\left(- \frac{2}{3} + \frac{2}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{6} + i \left(- \frac{2 \sqrt{3}}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{6}\right)\right) \left(- \frac{2}{3} + \frac{2}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{6} + \frac{2 \sqrt{3}}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}\right)\right) \left(- \frac{\sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{3} - \frac{2}{3} - \frac{4}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}\right)$$
=
-1
$$-1$$
-1
Respuesta rápida
[src]
________________ / ________________\
/ _____ | / _____ |
/ 43 3*\/ 177 | ___ / 43 3*\/ 177 |
3 / -- + --------- | ___ \/ 3 *3 / -- + --------- |
2 \/ 2 2 2 | 2*\/ 3 \/ 2 2 |
x1 = - - + --------------------- + ----------------------- + I*|- ----------------------- + ---------------------------|
3 6 ________________ | ________________ 6 |
/ _____ | / _____ |
/ 43 3*\/ 177 | / 43 3*\/ 177 |
3*3 / -- + --------- | 3*3 / -- + --------- |
\/ 2 2 \ \/ 2 2 /
$$x_{1} = - \frac{2}{3} + \frac{2}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{6} + i \left(- \frac{2 \sqrt{3}}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{6}\right)$$
________________ / ________________ \
/ _____ | / _____ |
/ 43 3*\/ 177 | ___ / 43 3*\/ 177 |
3 / -- + --------- | \/ 3 *3 / -- + --------- ___ |
2 \/ 2 2 2 | \/ 2 2 2*\/ 3 |
x2 = - - + --------------------- + ----------------------- + I*|- --------------------------- + -----------------------|
3 6 ________________ | 6 ________________|
/ _____ | / _____ |
/ 43 3*\/ 177 | / 43 3*\/ 177 |
3*3 / -- + --------- | 3*3 / -- + --------- |
\/ 2 2 \ \/ 2 2 /
$$x_{2} = - \frac{2}{3} + \frac{2}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}} + \frac{\sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{6} + \frac{2 \sqrt{3}}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}\right)$$
________________
/ _____
/ 43 3*\/ 177
3 / -- + ---------
2 4 \/ 2 2
x3 = - - - ----------------------- - ---------------------
3 ________________ 3
/ _____
/ 43 3*\/ 177
3*3 / -- + ---------
\/ 2 2
$$x_{3} = - \frac{\sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}{3} - \frac{2}{3} - \frac{4}{3 \sqrt[3]{\frac{3 \sqrt{177}}{2} + \frac{43}{2}}}$$
x3 = -(3*sqrt(177)/2 + 43/2)^(1/3)/3 - 2/3 - 4/(3*(3*sqrt(177)/2 + 43/2)^(1/3))
Respuesta numérica
[src]
x1 = -2.20556943040059
x2 = 0.102784715200295 - 0.665456951152813*i
x3 = 0.102784715200295 + 0.665456951152813*i
x3 = 0.102784715200295 + 0.665456951152813*i
Gráfico