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- La ecuación:
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Ecuación x^4-x^3+5*x^2=0
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Ecuación 3*x+5+(x+5)=(1-x)+4
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Ecuación (7-x)^2=(x+16)^2
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Ecuación 9x-7=6x+14
- Expresar {x} en función de y en la ecuación:
- -16*x-7*y=-5
- 7*x+19*y=12
- -5*x+5*y=-9
- -2*x-4*y=11
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Expresiones idénticas
- -x^ tres +x^ dos - dieciséis *x- veinte = cero
- menos x al cubo más x al cuadrado menos 16 multiplicar por x menos 20 es igual a 0
- menos x en el grado tres más x en el grado dos menos dieciséis multiplicar por x menos veinte es igual a cero
- -x3+x2-16*x-20=0
- -x³+x²-16*x-20=0
- -x en el grado 3+x en el grado 2-16*x-20=0
- -x^3+x^2-16x-20=0
- -x3+x2-16x-20=0
- -x^3+x^2-16*x-20=O
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Expresiones semejantes
- -x^3+x^2-16*x+20=0
- -x^3-x^2-16*x-20=0
- x^3+x^2-16*x-20=0
- -x^3+x^2+16*x-20=0
-x^3+x^2-16*x-20=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
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3 2 - x + x - 16*x - 20 = 0
$$\left(- 16 x + \left(- x^{3} + x^{2}\right)\right) - 20 = 0$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(- 16 x + \left(- x^{3} + x^{2}\right)\right) - 20 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - x^{2} + 16 x + 20 = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = 16$$
$$v = \frac{d}{a}$$
$$v = 20$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 1$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 16$$
$$x_{1} x_{2} x_{3} = 20$$
$$\left(- 16 x + \left(- x^{3} + x^{2}\right)\right) - 20 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - x^{2} + 16 x + 20 = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = 16$$
$$v = \frac{d}{a}$$
$$v = 20$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 1$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 16$$
$$x_{1} x_{2} x_{3} = 20$$
Suma y producto de raíces
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suma
__________________ / __________________ \ __________________ / __________________\ __________________
3 / ______ | ___ 3 / ______ ___ | 3 / ______ | ___ ___ 3 / ______ | 3 / ______
1 47 \/ 341 + 6*\/ 6114 |\/ 3 *\/ 341 + 6*\/ 6114 47*\/ 3 | 1 47 \/ 341 + 6*\/ 6114 | 47*\/ 3 \/ 3 *\/ 341 + 6*\/ 6114 | 1 \/ 341 + 6*\/ 6114 47
- - ----------------------- + --------------------- + I*|--------------------------- + -----------------------| + - - ----------------------- + --------------------- + I*|- ----------------------- - ---------------------------| + - - --------------------- + -----------------------
3 __________________ 6 | 6 __________________| 3 __________________ 6 | __________________ 6 | 3 3 __________________
3 / ______ | 3 / ______ | 3 / ______ | 3 / ______ | 3 / ______
6*\/ 341 + 6*\/ 6114 \ 6*\/ 341 + 6*\/ 6114 / 6*\/ 341 + 6*\/ 6114 \ 6*\/ 341 + 6*\/ 6114 / 3*\/ 341 + 6*\/ 6114
$$\left(- \frac{\sqrt[3]{341 + 6 \sqrt{6114}}}{3} + \frac{1}{3} + \frac{47}{3 \sqrt[3]{341 + 6 \sqrt{6114}}}\right) + \left(\left(- \frac{47}{6 \sqrt[3]{341 + 6 \sqrt{6114}}} + \frac{1}{3} + \frac{\sqrt[3]{341 + 6 \sqrt{6114}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{341 + 6 \sqrt{6114}}}{6} - \frac{47 \sqrt{3}}{6 \sqrt[3]{341 + 6 \sqrt{6114}}}\right)\right) + \left(- \frac{47}{6 \sqrt[3]{341 + 6 \sqrt{6114}}} + \frac{1}{3} + \frac{\sqrt[3]{341 + 6 \sqrt{6114}}}{6} + i \left(\frac{47 \sqrt{3}}{6 \sqrt[3]{341 + 6 \sqrt{6114}}} + \frac{\sqrt{3} \sqrt[3]{341 + 6 \sqrt{6114}}}{6}\right)\right)\right)$$
=
/ __________________\ / __________________ \
| ___ ___ 3 / ______ | | ___ 3 / ______ ___ |
| 47*\/ 3 \/ 3 *\/ 341 + 6*\/ 6114 | |\/ 3 *\/ 341 + 6*\/ 6114 47*\/ 3 |
1 + I*|- ----------------------- - ---------------------------| + I*|--------------------------- + -----------------------|
| __________________ 6 | | 6 __________________|
| 3 / ______ | | 3 / ______ |
\ 6*\/ 341 + 6*\/ 6114 / \ 6*\/ 341 + 6*\/ 6114 /
$$1 + i \left(- \frac{\sqrt{3} \sqrt[3]{341 + 6 \sqrt{6114}}}{6} - \frac{47 \sqrt{3}}{6 \sqrt[3]{341 + 6 \sqrt{6114}}}\right) + i \left(\frac{47 \sqrt{3}}{6 \sqrt[3]{341 + 6 \sqrt{6114}}} + \frac{\sqrt{3} \sqrt[3]{341 + 6 \sqrt{6114}}}{6}\right)$$
producto
/ __________________ / __________________ \\ / __________________ / __________________\\ / __________________ \ | 3 / ______ | ___ 3 / ______ ___ || | 3 / ______ | ___ ___ 3 / ______ || | 3 / ______ | |1 47 \/ 341 + 6*\/ 6114 |\/ 3 *\/ 341 + 6*\/ 6114 47*\/ 3 || |1 47 \/ 341 + 6*\/ 6114 | 47*\/ 3 \/ 3 *\/ 341 + 6*\/ 6114 || |1 \/ 341 + 6*\/ 6114 47 | |- - ----------------------- + --------------------- + I*|--------------------------- + -----------------------||*|- - ----------------------- + --------------------- + I*|- ----------------------- - ---------------------------||*|- - --------------------- + -----------------------| |3 __________________ 6 | 6 __________________|| |3 __________________ 6 | __________________ 6 || |3 3 __________________| | 3 / ______ | 3 / ______ || | 3 / ______ | 3 / ______ || | 3 / ______ | \ 6*\/ 341 + 6*\/ 6114 \ 6*\/ 341 + 6*\/ 6114 // \ 6*\/ 341 + 6*\/ 6114 \ 6*\/ 341 + 6*\/ 6114 // \ 3*\/ 341 + 6*\/ 6114 /
$$\left(- \frac{47}{6 \sqrt[3]{341 + 6 \sqrt{6114}}} + \frac{1}{3} + \frac{\sqrt[3]{341 + 6 \sqrt{6114}}}{6} + i \left(\frac{47 \sqrt{3}}{6 \sqrt[3]{341 + 6 \sqrt{6114}}} + \frac{\sqrt{3} \sqrt[3]{341 + 6 \sqrt{6114}}}{6}\right)\right) \left(- \frac{47}{6 \sqrt[3]{341 + 6 \sqrt{6114}}} + \frac{1}{3} + \frac{\sqrt[3]{341 + 6 \sqrt{6114}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{341 + 6 \sqrt{6114}}}{6} - \frac{47 \sqrt{3}}{6 \sqrt[3]{341 + 6 \sqrt{6114}}}\right)\right) \left(- \frac{\sqrt[3]{341 + 6 \sqrt{6114}}}{3} + \frac{1}{3} + \frac{47}{3 \sqrt[3]{341 + 6 \sqrt{6114}}}\right)$$
=
-20
$$-20$$
-20
Respuesta rápida
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__________________ / __________________ \
3 / ______ | ___ 3 / ______ ___ |
1 47 \/ 341 + 6*\/ 6114 |\/ 3 *\/ 341 + 6*\/ 6114 47*\/ 3 |
x1 = - - ----------------------- + --------------------- + I*|--------------------------- + -----------------------|
3 __________________ 6 | 6 __________________|
3 / ______ | 3 / ______ |
6*\/ 341 + 6*\/ 6114 \ 6*\/ 341 + 6*\/ 6114 /
$$x_{1} = - \frac{47}{6 \sqrt[3]{341 + 6 \sqrt{6114}}} + \frac{1}{3} + \frac{\sqrt[3]{341 + 6 \sqrt{6114}}}{6} + i \left(\frac{47 \sqrt{3}}{6 \sqrt[3]{341 + 6 \sqrt{6114}}} + \frac{\sqrt{3} \sqrt[3]{341 + 6 \sqrt{6114}}}{6}\right)$$
__________________ / __________________\
3 / ______ | ___ ___ 3 / ______ |
1 47 \/ 341 + 6*\/ 6114 | 47*\/ 3 \/ 3 *\/ 341 + 6*\/ 6114 |
x2 = - - ----------------------- + --------------------- + I*|- ----------------------- - ---------------------------|
3 __________________ 6 | __________________ 6 |
3 / ______ | 3 / ______ |
6*\/ 341 + 6*\/ 6114 \ 6*\/ 341 + 6*\/ 6114 /
$$x_{2} = - \frac{47}{6 \sqrt[3]{341 + 6 \sqrt{6114}}} + \frac{1}{3} + \frac{\sqrt[3]{341 + 6 \sqrt{6114}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{341 + 6 \sqrt{6114}}}{6} - \frac{47 \sqrt{3}}{6 \sqrt[3]{341 + 6 \sqrt{6114}}}\right)$$
__________________
3 / ______
1 \/ 341 + 6*\/ 6114 47
x3 = - - --------------------- + -----------------------
3 3 __________________
3 / ______
3*\/ 341 + 6*\/ 6114
$$x_{3} = - \frac{\sqrt[3]{341 + 6 \sqrt{6114}}}{3} + \frac{1}{3} + \frac{47}{3 \sqrt[3]{341 + 6 \sqrt{6114}}}$$
x3 = -(341 + 6*sqrt(6114))^(1/3)/3 + 1/3 + 47/(3*(341 + 6*sqrt(6114))^(1/3))
Respuesta numérica
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x1 = 1.0467663862312 - 4.1465198455534*i
x2 = 1.0467663862312 + 4.1465198455534*i
x3 = -1.0935327724624
x3 = -1.0935327724624