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- La ecuación:
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Ecuación x^(5/3)=32
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Ecuación (8/2)^(x+8)=6^x
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Ecuación 2+x=6
-
Ecuación 2x^2+13-13x=x+1
- Expresar {x} en función de y en la ecuación:
- -15*x-5*y=12
- -4*x-13*y=-14
- -16*x-14*y=-18
- -3*x+14*y=15
- Gráfico de la función y =:
-
x^3-x-1
- Factorizar el polinomio:
- x^3-x-1
-
Expresiones idénticas
- x^ tres -x- uno = cero
- x al cubo menos x menos 1 es igual a 0
- x en el grado tres menos x menos uno es igual a cero
- x3-x-1=0
- x³-x-1=0
- x en el grado 3-x-1=0
- x^3-x-1=O
-
Expresiones semejantes
- x^3-x+1=0
- x^3+x-1=0
x^3-x-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
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -1$$
$$v = \frac{d}{a}$$
$$v = -1$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -1$$
$$x_{1} x_{2} x_{3} = -1$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -1$$
$$v = \frac{d}{a}$$
$$v = -1$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -1$$
$$x_{1} x_{2} x_{3} = -1$$
Suma y producto de raíces
[src]
suma
____________ / ____________ \ ____________ / ____________ \
/ ____ | / ____ | / ____ | / ____ |
/ 1 \/ 69 | ___ / 1 \/ 69 | / 1 \/ 69 | ___ / 1 \/ 69 | ____________
3 / - + ------ | \/ 3 *3 / - + ------ ___ | 3 / - + ------ |\/ 3 *3 / - + ------ ___ | / ____
\/ 2 18 1 | \/ 2 18 \/ 3 | \/ 2 18 1 | \/ 2 18 \/ 3 | / 1 \/ 69 1
- ----------------- - ------------------- + I*|- ----------------------- + -------------------| + - ----------------- - ------------------- + I*|----------------------- - -------------------| + 3 / - + ------ + -------------------
2 ____________ | 2 ____________| 2 ____________ | 2 ____________| \/ 2 18 ____________
/ ____ | / ____ | / ____ | / ____ | / ____
/ 1 \/ 69 | / 1 \/ 69 | / 1 \/ 69 | / 1 \/ 69 | / 1 \/ 69
6*3 / - + ------ | 6*3 / - + ------ | 6*3 / - + ------ | 6*3 / - + ------ | 3*3 / - + ------
\/ 2 18 \ \/ 2 18 / \/ 2 18 \ \/ 2 18 / \/ 2 18
$$\left(\frac{1}{3 \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}} + \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}}{2} - \frac{1}{6 \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}}{2} + \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}}{2} - \frac{1}{6 \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}} + i \left(- \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}}{2}\right)\right)\right)$$
=
/ ____________ \ / ____________ \ | / ____ | | / ____ | | ___ / 1 \/ 69 | | ___ / 1 \/ 69 | |\/ 3 *3 / - + ------ ___ | | \/ 3 *3 / - + ------ ___ | | \/ 2 18 \/ 3 | | \/ 2 18 \/ 3 | I*|----------------------- - -------------------| + I*|- ----------------------- + -------------------| | 2 ____________| | 2 ____________| | / ____ | | / ____ | | / 1 \/ 69 | | / 1 \/ 69 | | 6*3 / - + ------ | | 6*3 / - + ------ | \ \/ 2 18 / \ \/ 2 18 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}}{2} + \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}}\right) + i \left(- \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}}{2}\right)$$
producto
/ ____________ / ____________ \\ / ____________ / ____________ \\ | / ____ | / ____ || | / ____ | / ____ || | / 1 \/ 69 | ___ / 1 \/ 69 || | / 1 \/ 69 | ___ / 1 \/ 69 || / ____________ \ | 3 / - + ------ | \/ 3 *3 / - + ------ ___ || | 3 / - + ------ |\/ 3 *3 / - + ------ ___ || | / ____ | | \/ 2 18 1 | \/ 2 18 \/ 3 || | \/ 2 18 1 | \/ 2 18 \/ 3 || | / 1 \/ 69 1 | |- ----------------- - ------------------- + I*|- ----------------------- + -------------------||*|- ----------------- - ------------------- + I*|----------------------- - -------------------||*|3 / - + ------ + -------------------| | 2 ____________ | 2 ____________|| | 2 ____________ | 2 ____________|| |\/ 2 18 ____________| | / ____ | / ____ || | / ____ | / ____ || | / ____ | | / 1 \/ 69 | / 1 \/ 69 || | / 1 \/ 69 | / 1 \/ 69 || | / 1 \/ 69 | | 6*3 / - + ------ | 6*3 / - + ------ || | 6*3 / - + ------ | 6*3 / - + ------ || | 3*3 / - + ------ | \ \/ 2 18 \ \/ 2 18 // \ \/ 2 18 \ \/ 2 18 // \ \/ 2 18 /
$$\left(- \frac{\sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}}{2} - \frac{1}{6 \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}} + i \left(- \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}}{2} - \frac{1}{6 \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}}{2} + \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}}\right)\right) \left(\frac{1}{3 \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}} + \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}\right)$$
=
1
$$1$$
1
Respuesta rápida
[src]
____________ / ____________ \
/ ____ | / ____ |
/ 1 \/ 69 | ___ / 1 \/ 69 |
3 / - + ------ | \/ 3 *3 / - + ------ ___ |
\/ 2 18 1 | \/ 2 18 \/ 3 |
x1 = - ----------------- - ------------------- + I*|- ----------------------- + -------------------|
2 ____________ | 2 ____________|
/ ____ | / ____ |
/ 1 \/ 69 | / 1 \/ 69 |
6*3 / - + ------ | 6*3 / - + ------ |
\/ 2 18 \ \/ 2 18 /
$$x_{1} = - \frac{\sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}}{2} - \frac{1}{6 \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}}{2} + \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}}\right)$$
____________ / ____________ \
/ ____ | / ____ |
/ 1 \/ 69 | ___ / 1 \/ 69 |
3 / - + ------ |\/ 3 *3 / - + ------ ___ |
\/ 2 18 1 | \/ 2 18 \/ 3 |
x2 = - ----------------- - ------------------- + I*|----------------------- - -------------------|
2 ____________ | 2 ____________|
/ ____ | / ____ |
/ 1 \/ 69 | / 1 \/ 69 |
6*3 / - + ------ | 6*3 / - + ------ |
\/ 2 18 \ \/ 2 18 /
$$x_{2} = - \frac{\sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}}{2} - \frac{1}{6 \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}} + i \left(- \frac{\sqrt{3}}{6 \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}}{2}\right)$$
____________
/ ____
/ 1 \/ 69 1
x3 = 3 / - + ------ + -------------------
\/ 2 18 ____________
/ ____
/ 1 \/ 69
3*3 / - + ------
\/ 2 18
$$x_{3} = \frac{1}{3 \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}} + \sqrt[3]{\frac{\sqrt{69}}{18} + \frac{1}{2}}$$
x3 = 1/(3*(sqrt(69)/18 + 1/2)^(1/3)) + (sqrt(69)/18 + 1/2)^(1/3)
Respuesta numérica
[src]
x1 = -0.662358978622373 - 0.562279512062301*i
x2 = 1.32471795724475
x3 = -0.662358978622373 + 0.562279512062301*i
x3 = -0.662358978622373 + 0.562279512062301*i
Gráfico