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- La ecuación:
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Ecuación 3x-2(3x+4)=10
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Ecuación x(x^2+2x+1)=6(x+1)
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Ecuación 4x+1=-3x+15
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Ecuación 6*3+x=72
- Expresar {x} en función de y en la ecuación:
- 7*x-1*y=0
- -18*x-6*y=-1
- 13*x-12*y=19
- 6*x+9*y=-9
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Expresiones idénticas
- dos *x^ tres - siete *x^ dos + cinco *x- diecinueve = cero
- 2 multiplicar por x al cubo menos 7 multiplicar por x al cuadrado más 5 multiplicar por x menos 19 es igual a 0
- dos multiplicar por x en el grado tres menos siete multiplicar por x en el grado dos más cinco multiplicar por x menos diecinueve es igual a cero
- 2*x3-7*x2+5*x-19=0
- 2*x³-7*x²+5*x-19=0
- 2*x en el grado 3-7*x en el grado 2+5*x-19=0
- 2x^3-7x^2+5x-19=0
- 2x3-7x2+5x-19=0
- 2*x^3-7*x^2+5*x-19=O
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Expresiones semejantes
- 2*x^3+7*x^2+5*x-19=0
- 2*x^3-7*x^2+5*x+19=0
- 2*x^3-7*x^2-5*x-19=0
2*x^3-7*x^2+5*x-19=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
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3 2 2*x - 7*x + 5*x - 19 = 0
$$\left(5 x + \left(2 x^{3} - 7 x^{2}\right)\right) - 19 = 0$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(5 x + \left(2 x^{3} - 7 x^{2}\right)\right) - 19 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - \frac{7 x^{2}}{2} + \frac{5 x}{2} - \frac{19}{2} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - \frac{7}{2}$$
$$q = \frac{c}{a}$$
$$q = \frac{5}{2}$$
$$v = \frac{d}{a}$$
$$v = - \frac{19}{2}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = \frac{7}{2}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = \frac{5}{2}$$
$$x_{1} x_{2} x_{3} = - \frac{19}{2}$$
$$\left(5 x + \left(2 x^{3} - 7 x^{2}\right)\right) - 19 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - \frac{7 x^{2}}{2} + \frac{5 x}{2} - \frac{19}{2} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - \frac{7}{2}$$
$$q = \frac{c}{a}$$
$$q = \frac{5}{2}$$
$$v = \frac{d}{a}$$
$$v = - \frac{19}{2}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = \frac{7}{2}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = \frac{5}{2}$$
$$x_{1} x_{2} x_{3} = - \frac{19}{2}$$
Suma y producto de raíces
[src]
suma
__________________ / __________________ \ __________________ / __________________ \
/ ________ | / ________ | / ________ | / ________ |
/ 527 \/ 122673 | ___ / 527 \/ 122673 | / 527 \/ 122673 | ___ / 527 \/ 122673 | __________________
3 / --- + ---------- | \/ 3 *3 / --- + ---------- ___ | 3 / --- + ---------- |\/ 3 *3 / --- + ---------- ___ | / ________
7 19 \/ 108 72 | \/ 108 72 19*\/ 3 | 7 19 \/ 108 72 | \/ 108 72 19*\/ 3 | 7 / 527 \/ 122673 19
- - -------------------------- - ----------------------- + I*|- ----------------------------- + --------------------------| + - - -------------------------- - ----------------------- + I*|----------------------------- - --------------------------| + - + 3 / --- + ---------- + --------------------------
6 __________________ 2 | 2 __________________| 6 __________________ 2 | 2 __________________| 6 \/ 108 72 __________________
/ ________ | / ________ | / ________ | / ________ | / ________
/ 527 \/ 122673 | / 527 \/ 122673 | / 527 \/ 122673 | / 527 \/ 122673 | / 527 \/ 122673
72*3 / --- + ---------- | 72*3 / --- + ---------- | 72*3 / --- + ---------- | 72*3 / --- + ---------- | 36*3 / --- + ----------
\/ 108 72 \ \/ 108 72 / \/ 108 72 \ \/ 108 72 / \/ 108 72
$$\left(\frac{19}{36 \sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}} + \frac{7}{6} + \sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}}{2} - \frac{19}{72 \sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}} + \frac{7}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}}{2} + \frac{19 \sqrt{3}}{72 \sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}}{2} - \frac{19}{72 \sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}} + \frac{7}{6} + i \left(- \frac{19 \sqrt{3}}{72 \sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}}{2}\right)\right)\right)$$
=
/ __________________ \ / __________________ \
| / ________ | | / ________ |
| ___ / 527 \/ 122673 | | ___ / 527 \/ 122673 |
|\/ 3 *3 / --- + ---------- ___ | | \/ 3 *3 / --- + ---------- ___ |
7 | \/ 108 72 19*\/ 3 | | \/ 108 72 19*\/ 3 |
- + I*|----------------------------- - --------------------------| + I*|- ----------------------------- + --------------------------|
2 | 2 __________________| | 2 __________________|
| / ________ | | / ________ |
| / 527 \/ 122673 | | / 527 \/ 122673 |
| 72*3 / --- + ---------- | | 72*3 / --- + ---------- |
\ \/ 108 72 / \ \/ 108 72 /
$$\frac{7}{2} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}}{2} + \frac{19 \sqrt{3}}{72 \sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}}\right) + i \left(- \frac{19 \sqrt{3}}{72 \sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}}{2}\right)$$
producto
/ __________________ / __________________ \\ / __________________ / __________________ \\ | / ________ | / ________ || | / ________ | / ________ || | / 527 \/ 122673 | ___ / 527 \/ 122673 || | / 527 \/ 122673 | ___ / 527 \/ 122673 || / __________________ \ | 3 / --- + ---------- | \/ 3 *3 / --- + ---------- ___ || | 3 / --- + ---------- |\/ 3 *3 / --- + ---------- ___ || | / ________ | |7 19 \/ 108 72 | \/ 108 72 19*\/ 3 || |7 19 \/ 108 72 | \/ 108 72 19*\/ 3 || |7 / 527 \/ 122673 19 | |- - -------------------------- - ----------------------- + I*|- ----------------------------- + --------------------------||*|- - -------------------------- - ----------------------- + I*|----------------------------- - --------------------------||*|- + 3 / --- + ---------- + --------------------------| |6 __________________ 2 | 2 __________________|| |6 __________________ 2 | 2 __________________|| |6 \/ 108 72 __________________| | / ________ | / ________ || | / ________ | / ________ || | / ________ | | / 527 \/ 122673 | / 527 \/ 122673 || | / 527 \/ 122673 | / 527 \/ 122673 || | / 527 \/ 122673 | | 72*3 / --- + ---------- | 72*3 / --- + ---------- || | 72*3 / --- + ---------- | 72*3 / --- + ---------- || | 36*3 / --- + ---------- | \ \/ 108 72 \ \/ 108 72 // \ \/ 108 72 \ \/ 108 72 // \ \/ 108 72 /
$$\left(- \frac{\sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}}{2} - \frac{19}{72 \sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}} + \frac{7}{6} + i \left(- \frac{19 \sqrt{3}}{72 \sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}}{2} - \frac{19}{72 \sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}} + \frac{7}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}}{2} + \frac{19 \sqrt{3}}{72 \sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}}\right)\right) \left(\frac{19}{36 \sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}} + \frac{7}{6} + \sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}\right)$$
=
19/2
$$\frac{19}{2}$$
19/2
Respuesta rápida
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__________________ / __________________ \
/ ________ | / ________ |
/ 527 \/ 122673 | ___ / 527 \/ 122673 |
3 / --- + ---------- | \/ 3 *3 / --- + ---------- ___ |
7 19 \/ 108 72 | \/ 108 72 19*\/ 3 |
x1 = - - -------------------------- - ----------------------- + I*|- ----------------------------- + --------------------------|
6 __________________ 2 | 2 __________________|
/ ________ | / ________ |
/ 527 \/ 122673 | / 527 \/ 122673 |
72*3 / --- + ---------- | 72*3 / --- + ---------- |
\/ 108 72 \ \/ 108 72 /
$$x_{1} = - \frac{\sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}}{2} - \frac{19}{72 \sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}} + \frac{7}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}}{2} + \frac{19 \sqrt{3}}{72 \sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}}\right)$$
__________________ / __________________ \
/ ________ | / ________ |
/ 527 \/ 122673 | ___ / 527 \/ 122673 |
3 / --- + ---------- |\/ 3 *3 / --- + ---------- ___ |
7 19 \/ 108 72 | \/ 108 72 19*\/ 3 |
x2 = - - -------------------------- - ----------------------- + I*|----------------------------- - --------------------------|
6 __________________ 2 | 2 __________________|
/ ________ | / ________ |
/ 527 \/ 122673 | / 527 \/ 122673 |
72*3 / --- + ---------- | 72*3 / --- + ---------- |
\/ 108 72 \ \/ 108 72 /
$$x_{2} = - \frac{\sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}}{2} - \frac{19}{72 \sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}} + \frac{7}{6} + i \left(- \frac{19 \sqrt{3}}{72 \sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}}{2}\right)$$
__________________
/ ________
7 / 527 \/ 122673 19
x3 = - + 3 / --- + ---------- + --------------------------
6 \/ 108 72 __________________
/ ________
/ 527 \/ 122673
36*3 / --- + ----------
\/ 108 72
$$x_{3} = \frac{19}{36 \sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}} + \frac{7}{6} + \sqrt[3]{\frac{\sqrt{122673}}{72} + \frac{527}{108}}$$
x3 = 19/(36*(sqrt(122673)/72 + 527/108)^(1/3)) + 7/6 + (sqrt(122673)/72 + 527/108)^(1/3)
Respuesta numérica
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x1 = -0.0248341974384107 - 1.63575352487962*i
x2 = 3.54966839487682
x3 = -0.0248341974384107 + 1.63575352487962*i
x3 = -0.0248341974384107 + 1.63575352487962*i