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Ecuación (-5*x-3)*(2*x-1)=0
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Ecuación x^3-x^2-5=0
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Expresiones idénticas
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- x3-x2-5=0
- x³-x²-5=0
- x en el grado 3-x en el grado 2-5=0
- x^3-x^2-5=O
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Expresiones semejantes
- x^3-x^2+5=0
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x^3-x^2-5=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
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = -5$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 1$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = -5$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = -5$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 1$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = -5$$
Respuesta rápida
[src]
________________ / ________________ \
/ ______ | / ______ |
/ 137 \/ 2085 | ___ / 137 \/ 2085 |
3 / --- + -------- | \/ 3 *3 / --- + -------- ___ |
1 \/ 54 18 1 | \/ 54 18 \/ 3 |
x1 = - - --------------------- - ------------------------ + I*|- --------------------------- + ------------------------|
3 2 ________________ | 2 ________________|
/ ______ | / ______ |
/ 137 \/ 2085 | / 137 \/ 2085 |
18*3 / --- + -------- | 18*3 / --- + -------- |
\/ 54 18 \ \/ 54 18 /
$$x_{1} = - \frac{\sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}{2} - \frac{1}{18 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}} + \frac{1}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}{2} + \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}\right)$$
________________ / ________________ \
/ ______ | / ______ |
/ 137 \/ 2085 | ___ / 137 \/ 2085 |
3 / --- + -------- |\/ 3 *3 / --- + -------- ___ |
1 \/ 54 18 1 | \/ 54 18 \/ 3 |
x2 = - - --------------------- - ------------------------ + I*|--------------------------- - ------------------------|
3 2 ________________ | 2 ________________|
/ ______ | / ______ |
/ 137 \/ 2085 | / 137 \/ 2085 |
18*3 / --- + -------- | 18*3 / --- + -------- |
\/ 54 18 \ \/ 54 18 /
$$x_{2} = - \frac{\sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}{2} - \frac{1}{18 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}} + \frac{1}{3} + i \left(- \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}{2}\right)$$
________________
/ ______
1 / 137 \/ 2085 1
x3 = - + 3 / --- + -------- + -----------------------
3 \/ 54 18 ________________
/ ______
/ 137 \/ 2085
9*3 / --- + --------
\/ 54 18
$$x_{3} = \frac{1}{9 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}} + \frac{1}{3} + \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}$$
x3 = 1/(9*(sqrt(2085)/18 + 137/54)^(1/3)) + 1/3 + (sqrt(2085)/18 + 137/54)^(1/3)
Suma y producto de raíces
[src]
suma
________________ / ________________ \ ________________ / ________________ \
/ ______ | / ______ | / ______ | / ______ |
/ 137 \/ 2085 | ___ / 137 \/ 2085 | / 137 \/ 2085 | ___ / 137 \/ 2085 | ________________
3 / --- + -------- | \/ 3 *3 / --- + -------- ___ | 3 / --- + -------- |\/ 3 *3 / --- + -------- ___ | / ______
1 \/ 54 18 1 | \/ 54 18 \/ 3 | 1 \/ 54 18 1 | \/ 54 18 \/ 3 | 1 / 137 \/ 2085 1
- - --------------------- - ------------------------ + I*|- --------------------------- + ------------------------| + - - --------------------- - ------------------------ + I*|--------------------------- - ------------------------| + - + 3 / --- + -------- + -----------------------
3 2 ________________ | 2 ________________| 3 2 ________________ | 2 ________________| 3 \/ 54 18 ________________
/ ______ | / ______ | / ______ | / ______ | / ______
/ 137 \/ 2085 | / 137 \/ 2085 | / 137 \/ 2085 | / 137 \/ 2085 | / 137 \/ 2085
18*3 / --- + -------- | 18*3 / --- + -------- | 18*3 / --- + -------- | 18*3 / --- + -------- | 9*3 / --- + --------
\/ 54 18 \ \/ 54 18 / \/ 54 18 \ \/ 54 18 / \/ 54 18
$$\left(\frac{1}{9 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}} + \frac{1}{3} + \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}{2} - \frac{1}{18 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}} + \frac{1}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}{2} + \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}{2} - \frac{1}{18 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}} + \frac{1}{3} + i \left(- \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}{2}\right)\right)\right)$$
=
/ ________________ \ / ________________ \
| / ______ | | / ______ |
| ___ / 137 \/ 2085 | | ___ / 137 \/ 2085 |
|\/ 3 *3 / --- + -------- ___ | | \/ 3 *3 / --- + -------- ___ |
| \/ 54 18 \/ 3 | | \/ 54 18 \/ 3 |
1 + I*|--------------------------- - ------------------------| + I*|- --------------------------- + ------------------------|
| 2 ________________| | 2 ________________|
| / ______ | | / ______ |
| / 137 \/ 2085 | | / 137 \/ 2085 |
| 18*3 / --- + -------- | | 18*3 / --- + -------- |
\ \/ 54 18 / \ \/ 54 18 /
$$1 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}{2} + \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}\right) + i \left(- \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}{2}\right)$$
producto
/ ________________ / ________________ \\ / ________________ / ________________ \\ | / ______ | / ______ || | / ______ | / ______ || | / 137 \/ 2085 | ___ / 137 \/ 2085 || | / 137 \/ 2085 | ___ / 137 \/ 2085 || / ________________ \ | 3 / --- + -------- | \/ 3 *3 / --- + -------- ___ || | 3 / --- + -------- |\/ 3 *3 / --- + -------- ___ || | / ______ | |1 \/ 54 18 1 | \/ 54 18 \/ 3 || |1 \/ 54 18 1 | \/ 54 18 \/ 3 || |1 / 137 \/ 2085 1 | |- - --------------------- - ------------------------ + I*|- --------------------------- + ------------------------||*|- - --------------------- - ------------------------ + I*|--------------------------- - ------------------------||*|- + 3 / --- + -------- + -----------------------| |3 2 ________________ | 2 ________________|| |3 2 ________________ | 2 ________________|| |3 \/ 54 18 ________________| | / ______ | / ______ || | / ______ | / ______ || | / ______ | | / 137 \/ 2085 | / 137 \/ 2085 || | / 137 \/ 2085 | / 137 \/ 2085 || | / 137 \/ 2085 | | 18*3 / --- + -------- | 18*3 / --- + -------- || | 18*3 / --- + -------- | 18*3 / --- + -------- || | 9*3 / --- + -------- | \ \/ 54 18 \ \/ 54 18 // \ \/ 54 18 \ \/ 54 18 // \ \/ 54 18 /
$$\left(- \frac{\sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}{2} - \frac{1}{18 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}} + \frac{1}{3} + i \left(- \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}{2} - \frac{1}{18 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}} + \frac{1}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}{2} + \frac{\sqrt{3}}{18 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}}\right)\right) \left(\frac{1}{9 \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}} + \frac{1}{3} + \sqrt[3]{\frac{\sqrt{2085}}{18} + \frac{137}{54}}\right)$$
=
5
$$5$$
5
Respuesta numérica
[src]
x1 = -0.558171649312106 - 1.43213479425353*i
x2 = -0.558171649312106 + 1.43213479425353*i
x3 = 2.11634329862421
x3 = 2.11634329862421
Gráfico