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- La ecuación:
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Ecuación x^2+5x=36
-
Ecuación 3*x-6=x+4
-
Ecuación 9^x-8*3^x-9=0
-
Ecuación x^3+4=0
- Expresar {x} en función de y en la ecuación:
- -2*x-19*y=3
- 6*x+17*y=-11
- -11*x-7*y=-2
- -19*x+14*y=20
- Factorizar el polinomio:
- x^3+2*x-11
-
Expresiones idénticas
- x^ tres + dos *x- once = cero
- x al cubo más 2 multiplicar por x menos 11 es igual a 0
- x en el grado tres más dos multiplicar por x menos once es igual a cero
- x3+2*x-11=0
- x³+2*x-11=0
- x en el grado 3+2*x-11=0
- x^3+2x-11=0
- x3+2x-11=0
- x^3+2*x-11=O
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Expresiones semejantes
- x^3+2*x+11=0
- x^3-2*x-11=0
x^3+2*x-11=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
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 2$$
$$v = \frac{d}{a}$$
$$v = -11$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 2$$
$$x_{1} x_{2} x_{3} = -11$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = 2$$
$$v = \frac{d}{a}$$
$$v = -11$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 2$$
$$x_{1} x_{2} x_{3} = -11$$
Respuesta rápida
[src]
_______________ / _______________ \
/ ______ | / ______ |
/ 11 \/ 9897 | ___ / 11 \/ 9897 |
3 / -- + -------- | \/ 3 *3 / -- + -------- ___ |
\/ 2 18 1 | \/ 2 18 \/ 3 |
x1 = - -------------------- + ---------------------- + I*|- -------------------------- - ----------------------|
2 _______________ | 2 _______________|
/ ______ | / ______ |
/ 11 \/ 9897 | / 11 \/ 9897 |
3*3 / -- + -------- | 3*3 / -- + -------- |
\/ 2 18 \ \/ 2 18 /
$$x_{1} = - \frac{\sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}}{2} + \frac{1}{3 \sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}}{2} - \frac{\sqrt{3}}{3 \sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}}\right)$$
_______________ / _______________ \
/ ______ | / ______ |
/ 11 \/ 9897 | ___ / 11 \/ 9897 |
3 / -- + -------- |\/ 3 *3 / -- + -------- ___ |
\/ 2 18 1 | \/ 2 18 \/ 3 |
x2 = - -------------------- + ---------------------- + I*|-------------------------- + ----------------------|
2 _______________ | 2 _______________|
/ ______ | / ______ |
/ 11 \/ 9897 | / 11 \/ 9897 |
3*3 / -- + -------- | 3*3 / -- + -------- |
\/ 2 18 \ \/ 2 18 /
$$x_{2} = - \frac{\sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}}{2} + \frac{1}{3 \sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}} + i \left(\frac{\sqrt{3}}{3 \sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}} + \frac{\sqrt{3} \sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}}{2}\right)$$
_______________
/ ______
/ 11 \/ 9897 2
x3 = 3 / -- + -------- - ----------------------
\/ 2 18 _______________
/ ______
/ 11 \/ 9897
3*3 / -- + --------
\/ 2 18
$$x_{3} = - \frac{2}{3 \sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}} + \sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}$$
x3 = -2/(3*(11/2 + sqrt(9897)/18)^(1/3)) + (11/2 + sqrt(9897)/18)^(1/3)
Suma y producto de raíces
[src]
suma
_______________ / _______________ \ _______________ / _______________ \
/ ______ | / ______ | / ______ | / ______ |
/ 11 \/ 9897 | ___ / 11 \/ 9897 | / 11 \/ 9897 | ___ / 11 \/ 9897 | _______________
3 / -- + -------- | \/ 3 *3 / -- + -------- ___ | 3 / -- + -------- |\/ 3 *3 / -- + -------- ___ | / ______
\/ 2 18 1 | \/ 2 18 \/ 3 | \/ 2 18 1 | \/ 2 18 \/ 3 | / 11 \/ 9897 2
- -------------------- + ---------------------- + I*|- -------------------------- - ----------------------| + - -------------------- + ---------------------- + I*|-------------------------- + ----------------------| + 3 / -- + -------- - ----------------------
2 _______________ | 2 _______________| 2 _______________ | 2 _______________| \/ 2 18 _______________
/ ______ | / ______ | / ______ | / ______ | / ______
/ 11 \/ 9897 | / 11 \/ 9897 | / 11 \/ 9897 | / 11 \/ 9897 | / 11 \/ 9897
3*3 / -- + -------- | 3*3 / -- + -------- | 3*3 / -- + -------- | 3*3 / -- + -------- | 3*3 / -- + --------
\/ 2 18 \ \/ 2 18 / \/ 2 18 \ \/ 2 18 / \/ 2 18
$$\left(- \frac{2}{3 \sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}} + \sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}}{2} + \frac{1}{3 \sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}}{2} - \frac{\sqrt{3}}{3 \sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}}{2} + \frac{1}{3 \sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}} + i \left(\frac{\sqrt{3}}{3 \sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}} + \frac{\sqrt{3} \sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}}{2}\right)\right)\right)$$
=
/ _______________ \ / _______________ \ | / ______ | | / ______ | | ___ / 11 \/ 9897 | | ___ / 11 \/ 9897 | |\/ 3 *3 / -- + -------- ___ | | \/ 3 *3 / -- + -------- ___ | | \/ 2 18 \/ 3 | | \/ 2 18 \/ 3 | I*|-------------------------- + ----------------------| + I*|- -------------------------- - ----------------------| | 2 _______________| | 2 _______________| | / ______ | | / ______ | | / 11 \/ 9897 | | / 11 \/ 9897 | | 3*3 / -- + -------- | | 3*3 / -- + -------- | \ \/ 2 18 / \ \/ 2 18 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}}{2} - \frac{\sqrt{3}}{3 \sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}}\right) + i \left(\frac{\sqrt{3}}{3 \sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}} + \frac{\sqrt{3} \sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}}{2}\right)$$
producto
/ _______________ / _______________ \\ / _______________ / _______________ \\ | / ______ | / ______ || | / ______ | / ______ || | / 11 \/ 9897 | ___ / 11 \/ 9897 || | / 11 \/ 9897 | ___ / 11 \/ 9897 || / _______________ \ | 3 / -- + -------- | \/ 3 *3 / -- + -------- ___ || | 3 / -- + -------- |\/ 3 *3 / -- + -------- ___ || | / ______ | | \/ 2 18 1 | \/ 2 18 \/ 3 || | \/ 2 18 1 | \/ 2 18 \/ 3 || | / 11 \/ 9897 2 | |- -------------------- + ---------------------- + I*|- -------------------------- - ----------------------||*|- -------------------- + ---------------------- + I*|-------------------------- + ----------------------||*|3 / -- + -------- - ----------------------| | 2 _______________ | 2 _______________|| | 2 _______________ | 2 _______________|| |\/ 2 18 _______________| | / ______ | / ______ || | / ______ | / ______ || | / ______ | | / 11 \/ 9897 | / 11 \/ 9897 || | / 11 \/ 9897 | / 11 \/ 9897 || | / 11 \/ 9897 | | 3*3 / -- + -------- | 3*3 / -- + -------- || | 3*3 / -- + -------- | 3*3 / -- + -------- || | 3*3 / -- + -------- | \ \/ 2 18 \ \/ 2 18 // \ \/ 2 18 \ \/ 2 18 // \ \/ 2 18 /
$$\left(- \frac{\sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}}{2} + \frac{1}{3 \sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}} + i \left(\frac{\sqrt{3}}{3 \sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}} + \frac{\sqrt{3} \sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}}{2} + \frac{1}{3 \sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}}{2} - \frac{\sqrt{3}}{3 \sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}}\right)\right) \left(- \frac{2}{3 \sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}} + \sqrt[3]{\frac{11}{2} + \frac{\sqrt{9897}}{18}}\right)$$
=
11
$$11$$
11
Respuesta numérica
[src]
x1 = -0.963135157148397 + 2.1869814797583*i
x2 = -0.963135157148397 - 2.1869814797583*i
x3 = 1.92627031429679
x3 = 1.92627031429679