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- La ecuación:
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Ecuación x^2+11=0
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Ecuación x^3-x^2-x-1=0
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Ecuación 3x-2(3x+4)=10
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Ecuación 5x+12=8x+30
- Expresar {x} en función de y en la ecuación:
- 7*x-1*y=0
- -11*x-3*y=14
- 2*x-15*y=-1
- -18*x-6*y=-1
- Gráfico de la función y =:
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x^3-2*x-5
- Forma canónica:
- x^3-2*x-5
-
Expresiones idénticas
- x^ tres - dos *x- cinco = cero
- x al cubo menos 2 multiplicar por x menos 5 es igual a 0
- x en el grado tres menos dos multiplicar por x menos cinco es igual a cero
- x3-2*x-5=0
- x³-2*x-5=0
- x en el grado 3-2*x-5=0
- x^3-2x-5=0
- x3-2x-5=0
- x^3-2*x-5=O
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Expresiones semejantes
- x^3+2*x-5=0
- x^3-2*x+5=0
x^3-2*x-5=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
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -2$$
$$v = \frac{d}{a}$$
$$v = -5$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -2$$
$$x_{1} x_{2} x_{3} = -5$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -2$$
$$v = \frac{d}{a}$$
$$v = -5$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -2$$
$$x_{1} x_{2} x_{3} = -5$$
Respuesta rápida
[src]
______________ / ______________ \
/ ______ | / ______ |
/ 5 \/ 1929 | ___ / 5 \/ 1929 |
3 / - + -------- | \/ 3 *3 / - + -------- ___ |
\/ 2 18 1 | \/ 2 18 \/ 3 |
x1 = - ------------------- - --------------------- + I*|- ------------------------- + ---------------------|
2 ______________ | 2 ______________|
/ ______ | / ______ |
/ 5 \/ 1929 | / 5 \/ 1929 |
3*3 / - + -------- | 3*3 / - + -------- |
\/ 2 18 \ \/ 2 18 /
$$x_{1} = - \frac{\sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}{2} - \frac{1}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}{2} + \frac{\sqrt{3}}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}\right)$$
______________ / ______________ \
/ ______ | / ______ |
/ 5 \/ 1929 | ___ / 5 \/ 1929 |
3 / - + -------- |\/ 3 *3 / - + -------- ___ |
\/ 2 18 1 | \/ 2 18 \/ 3 |
x2 = - ------------------- - --------------------- + I*|------------------------- - ---------------------|
2 ______________ | 2 ______________|
/ ______ | / ______ |
/ 5 \/ 1929 | / 5 \/ 1929 |
3*3 / - + -------- | 3*3 / - + -------- |
\/ 2 18 \ \/ 2 18 /
$$x_{2} = - \frac{\sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}{2} - \frac{1}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}} + i \left(- \frac{\sqrt{3}}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}{2}\right)$$
______________
/ ______
/ 5 \/ 1929 2
x3 = 3 / - + -------- + ---------------------
\/ 2 18 ______________
/ ______
/ 5 \/ 1929
3*3 / - + --------
\/ 2 18
$$x_{3} = \frac{2}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}} + \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}$$
x3 = 2/(3*(sqrt(1929)/18 + 5/2)^(1/3)) + (sqrt(1929)/18 + 5/2)^(1/3)
Suma y producto de raíces
[src]
suma
______________ / ______________ \ ______________ / ______________ \
/ ______ | / ______ | / ______ | / ______ |
/ 5 \/ 1929 | ___ / 5 \/ 1929 | / 5 \/ 1929 | ___ / 5 \/ 1929 | ______________
3 / - + -------- | \/ 3 *3 / - + -------- ___ | 3 / - + -------- |\/ 3 *3 / - + -------- ___ | / ______
\/ 2 18 1 | \/ 2 18 \/ 3 | \/ 2 18 1 | \/ 2 18 \/ 3 | / 5 \/ 1929 2
- ------------------- - --------------------- + I*|- ------------------------- + ---------------------| + - ------------------- - --------------------- + I*|------------------------- - ---------------------| + 3 / - + -------- + ---------------------
2 ______________ | 2 ______________| 2 ______________ | 2 ______________| \/ 2 18 ______________
/ ______ | / ______ | / ______ | / ______ | / ______
/ 5 \/ 1929 | / 5 \/ 1929 | / 5 \/ 1929 | / 5 \/ 1929 | / 5 \/ 1929
3*3 / - + -------- | 3*3 / - + -------- | 3*3 / - + -------- | 3*3 / - + -------- | 3*3 / - + --------
\/ 2 18 \ \/ 2 18 / \/ 2 18 \ \/ 2 18 / \/ 2 18
$$\left(\frac{2}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}} + \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}{2} - \frac{1}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}{2} + \frac{\sqrt{3}}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}{2} - \frac{1}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}} + i \left(- \frac{\sqrt{3}}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}{2}\right)\right)\right)$$
=
/ ______________ \ / ______________ \ | / ______ | | / ______ | | ___ / 5 \/ 1929 | | ___ / 5 \/ 1929 | |\/ 3 *3 / - + -------- ___ | | \/ 3 *3 / - + -------- ___ | | \/ 2 18 \/ 3 | | \/ 2 18 \/ 3 | I*|------------------------- - ---------------------| + I*|- ------------------------- + ---------------------| | 2 ______________| | 2 ______________| | / ______ | | / ______ | | / 5 \/ 1929 | | / 5 \/ 1929 | | 3*3 / - + -------- | | 3*3 / - + -------- | \ \/ 2 18 / \ \/ 2 18 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}{2} + \frac{\sqrt{3}}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}\right) + i \left(- \frac{\sqrt{3}}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}{2}\right)$$
producto
/ ______________ / ______________ \\ / ______________ / ______________ \\ | / ______ | / ______ || | / ______ | / ______ || | / 5 \/ 1929 | ___ / 5 \/ 1929 || | / 5 \/ 1929 | ___ / 5 \/ 1929 || / ______________ \ | 3 / - + -------- | \/ 3 *3 / - + -------- ___ || | 3 / - + -------- |\/ 3 *3 / - + -------- ___ || | / ______ | | \/ 2 18 1 | \/ 2 18 \/ 3 || | \/ 2 18 1 | \/ 2 18 \/ 3 || | / 5 \/ 1929 2 | |- ------------------- - --------------------- + I*|- ------------------------- + ---------------------||*|- ------------------- - --------------------- + I*|------------------------- - ---------------------||*|3 / - + -------- + ---------------------| | 2 ______________ | 2 ______________|| | 2 ______________ | 2 ______________|| |\/ 2 18 ______________| | / ______ | / ______ || | / ______ | / ______ || | / ______ | | / 5 \/ 1929 | / 5 \/ 1929 || | / 5 \/ 1929 | / 5 \/ 1929 || | / 5 \/ 1929 | | 3*3 / - + -------- | 3*3 / - + -------- || | 3*3 / - + -------- | 3*3 / - + -------- || | 3*3 / - + -------- | \ \/ 2 18 \ \/ 2 18 // \ \/ 2 18 \ \/ 2 18 // \ \/ 2 18 /
$$\left(- \frac{\sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}{2} - \frac{1}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}} + i \left(- \frac{\sqrt{3}}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}{2} - \frac{1}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}{2} + \frac{\sqrt{3}}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}}\right)\right) \left(\frac{2}{3 \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}} + \sqrt[3]{\frac{\sqrt{1929}}{18} + \frac{5}{2}}\right)$$
=
5
$$5$$
5
Respuesta numérica
[src]
x1 = 2.09455148154233
x2 = -1.04727574077116 - 1.13593988908893*i
x3 = -1.04727574077116 + 1.13593988908893*i
x3 = -1.04727574077116 + 1.13593988908893*i
Gráfico