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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
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Ecuación x^2=2x+8
- Expresar {x} en función de y en la ecuación:
- -11*x-14*y=-3
- -9*x-1*y=6
- -2*x+2*y=4
- -11*x-3*y=14
- Derivada de:
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x^3+3x+1
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Expresiones idénticas
- x^ tres +3x+ uno = cero
- x al cubo más 3x más 1 es igual a 0
- x en el grado tres más 3x más uno es igual a cero
- x3+3x+1=0
- x³+3x+1=0
- x en el grado 3+3x+1=0
- x^3+3x+1=O
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Expresiones semejantes
- x^3-3x+1=0
- x^3+3x-1=0
x^3+3x+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
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 3$$
$$v = \frac{d}{a}$$
$$v = 1$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 3$$
$$x_{1} x_{2} x_{3} = 1$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 3$$
$$v = \frac{d}{a}$$
$$v = 1$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 3$$
$$x_{1} x_{2} x_{3} = 1$$
Respuesta rápida
[src]
_______________ / _______________ \
/ ___ | / ___ |
/ 27 27*\/ 5 | ___ / 27 27*\/ 5 |
3 / -- + -------- |\/ 3 *3 / -- + -------- ___ |
3 \/ 2 2 | \/ 2 2 3*\/ 3 |
x1 = - ---------------------- + -------------------- + I*|-------------------------- + ----------------------|
_______________ 6 | 6 _______________|
/ ___ | / ___ |
/ 27 27*\/ 5 | / 27 27*\/ 5 |
2*3 / -- + -------- | 2*3 / -- + -------- |
\/ 2 2 \ \/ 2 2 /
$$x_{1} = - \frac{3}{2 \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}} + \frac{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}}{6} + i \left(\frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}}{6}\right)$$
_______________ / _______________\
/ ___ | / ___ |
/ 27 27*\/ 5 | ___ / 27 27*\/ 5 |
3 / -- + -------- | ___ \/ 3 *3 / -- + -------- |
3 \/ 2 2 | 3*\/ 3 \/ 2 2 |
x2 = - ---------------------- + -------------------- + I*|- ---------------------- - --------------------------|
_______________ 6 | _______________ 6 |
/ ___ | / ___ |
/ 27 27*\/ 5 | / 27 27*\/ 5 |
2*3 / -- + -------- | 2*3 / -- + -------- |
\/ 2 2 \ \/ 2 2 /
$$x_{2} = - \frac{3}{2 \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}} + \frac{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}}{6} - \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}}\right)$$
_______________
/ ___
/ 27 27*\/ 5
3 / -- + --------
3 \/ 2 2
x3 = -------------------- - --------------------
_______________ 3
/ ___
/ 27 27*\/ 5
3 / -- + --------
\/ 2 2
$$x_{3} = - \frac{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}}{3} + \frac{3}{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}}$$
x3 = -(27/2 + 27*sqrt(5)/2)^(1/3)/3 + 3/(27/2 + 27*sqrt(5)/2)^(1/3)
Suma y producto de raíces
[src]
suma
_______________ / _______________ \ _______________ / _______________\ _______________
/ ___ | / ___ | / ___ | / ___ | / ___
/ 27 27*\/ 5 | ___ / 27 27*\/ 5 | / 27 27*\/ 5 | ___ / 27 27*\/ 5 | / 27 27*\/ 5
3 / -- + -------- |\/ 3 *3 / -- + -------- ___ | 3 / -- + -------- | ___ \/ 3 *3 / -- + -------- | 3 / -- + --------
3 \/ 2 2 | \/ 2 2 3*\/ 3 | 3 \/ 2 2 | 3*\/ 3 \/ 2 2 | 3 \/ 2 2
- ---------------------- + -------------------- + I*|-------------------------- + ----------------------| + - ---------------------- + -------------------- + I*|- ---------------------- - --------------------------| + -------------------- - --------------------
_______________ 6 | 6 _______________| _______________ 6 | _______________ 6 | _______________ 3
/ ___ | / ___ | / ___ | / ___ | / ___
/ 27 27*\/ 5 | / 27 27*\/ 5 | / 27 27*\/ 5 | / 27 27*\/ 5 | / 27 27*\/ 5
2*3 / -- + -------- | 2*3 / -- + -------- | 2*3 / -- + -------- | 2*3 / -- + -------- | 3 / -- + --------
\/ 2 2 \ \/ 2 2 / \/ 2 2 \ \/ 2 2 / \/ 2 2
$$\left(- \frac{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}}{3} + \frac{3}{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}}\right) + \left(\left(- \frac{3}{2 \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}} + \frac{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}}{6} - \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}}\right)\right) + \left(- \frac{3}{2 \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}} + \frac{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}}{6} + i \left(\frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}}{6}\right)\right)\right)$$
=
/ _______________\ / _______________ \ | / ___ | | / ___ | | ___ / 27 27*\/ 5 | | ___ / 27 27*\/ 5 | | ___ \/ 3 *3 / -- + -------- | |\/ 3 *3 / -- + -------- ___ | | 3*\/ 3 \/ 2 2 | | \/ 2 2 3*\/ 3 | I*|- ---------------------- - --------------------------| + I*|-------------------------- + ----------------------| | _______________ 6 | | 6 _______________| | / ___ | | / ___ | | / 27 27*\/ 5 | | / 27 27*\/ 5 | | 2*3 / -- + -------- | | 2*3 / -- + -------- | \ \/ 2 2 / \ \/ 2 2 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}}{6} - \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}}\right) + i \left(\frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}}{6}\right)$$
producto
/ _______________ / _______________ \\ / _______________ / _______________\\ / _______________\ | / ___ | / ___ || | / ___ | / ___ || | / ___ | | / 27 27*\/ 5 | ___ / 27 27*\/ 5 || | / 27 27*\/ 5 | ___ / 27 27*\/ 5 || | / 27 27*\/ 5 | | 3 / -- + -------- |\/ 3 *3 / -- + -------- ___ || | 3 / -- + -------- | ___ \/ 3 *3 / -- + -------- || | 3 / -- + -------- | | 3 \/ 2 2 | \/ 2 2 3*\/ 3 || | 3 \/ 2 2 | 3*\/ 3 \/ 2 2 || | 3 \/ 2 2 | |- ---------------------- + -------------------- + I*|-------------------------- + ----------------------||*|- ---------------------- + -------------------- + I*|- ---------------------- - --------------------------||*|-------------------- - --------------------| | _______________ 6 | 6 _______________|| | _______________ 6 | _______________ 6 || | _______________ 3 | | / ___ | / ___ || | / ___ | / ___ || | / ___ | | / 27 27*\/ 5 | / 27 27*\/ 5 || | / 27 27*\/ 5 | / 27 27*\/ 5 || | / 27 27*\/ 5 | | 2*3 / -- + -------- | 2*3 / -- + -------- || | 2*3 / -- + -------- | 2*3 / -- + -------- || |3 / -- + -------- | \ \/ 2 2 \ \/ 2 2 // \ \/ 2 2 \ \/ 2 2 // \\/ 2 2 /
$$\left(- \frac{3}{2 \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}} + \frac{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}}{6} + i \left(\frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}}{6}\right)\right) \left(- \frac{3}{2 \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}} + \frac{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}}{6} - \frac{3 \sqrt{3}}{2 \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}}\right)\right) \left(- \frac{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}}{3} + \frac{3}{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{5}}{2}}}\right)$$
=
-1
$$-1$$
-1
Respuesta numérica
[src]
x1 = 0.161092677313043 + 1.75438095978372*i
x2 = -0.322185354626086
x3 = 0.161092677313043 - 1.75438095978372*i
x3 = 0.161092677313043 - 1.75438095978372*i
Gráfico