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Ecuación 13x+10=6x-4
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Expresiones idénticas
- x^ tres -3x- nueve = cero
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- x³-3x-9=0
- x en el grado 3-3x-9=0
- x^3-3x-9=O
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Expresiones semejantes
- x^3+3x-9=0
- x^3-3x+9=0
x^3-3x-9=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
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -3$$
$$v = \frac{d}{a}$$
$$v = -9$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -3$$
$$x_{1} x_{2} x_{3} = -9$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -3$$
$$v = \frac{d}{a}$$
$$v = -9$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -3$$
$$x_{1} x_{2} x_{3} = -9$$
Suma y producto de raíces
[src]
suma
____________ / ____________\ ____________ / ____________ \
/ ____ | / ____ | / ____ | / ____ |
/ 9 \/ 77 | ___ / 9 \/ 77 | / 9 \/ 77 | ___ / 9 \/ 77 | ____________
3 / - + ------ | ___ \/ 3 *3 / - + ------ | 3 / - + ------ |\/ 3 *3 / - + ------ ___ | / ____
1 \/ 2 2 | \/ 3 \/ 2 2 | 1 \/ 2 2 | \/ 2 2 \/ 3 | 1 / 9 \/ 77
- ------------------- - ----------------- + I*|------------------- - -----------------------| + - ------------------- - ----------------- + I*|----------------------- - -------------------| + ----------------- + 3 / - + ------
____________ 2 | ____________ 2 | ____________ 2 | 2 ____________| ____________ \/ 2 2
/ ____ | / ____ | / ____ | / ____ | / ____
/ 9 \/ 77 | / 9 \/ 77 | / 9 \/ 77 | / 9 \/ 77 | / 9 \/ 77
2*3 / - + ------ |2*3 / - + ------ | 2*3 / - + ------ | 2*3 / - + ------ | 3 / - + ------
\/ 2 2 \ \/ 2 2 / \/ 2 2 \ \/ 2 2 / \/ 2 2
$$\left(\frac{1}{\sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}} + \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}{2} - \frac{1}{2 \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}{2} + \frac{\sqrt{3}}{2 \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}{2} - \frac{1}{2 \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}} + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}{2}\right)\right)\right)$$
=
/ ____________\ / ____________ \ | / ____ | | / ____ | | ___ / 9 \/ 77 | | ___ / 9 \/ 77 | | ___ \/ 3 *3 / - + ------ | |\/ 3 *3 / - + ------ ___ | | \/ 3 \/ 2 2 | | \/ 2 2 \/ 3 | I*|------------------- - -----------------------| + I*|----------------------- - -------------------| | ____________ 2 | | 2 ____________| | / ____ | | / ____ | | / 9 \/ 77 | | / 9 \/ 77 | |2*3 / - + ------ | | 2*3 / - + ------ | \ \/ 2 2 / \ \/ 2 2 /
$$i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}{2} + \frac{\sqrt{3}}{2 \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}\right) + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}{2}\right)$$
producto
/ ____________ / ____________\\ / ____________ / ____________ \\ | / ____ | / ____ || | / ____ | / ____ || | / 9 \/ 77 | ___ / 9 \/ 77 || | / 9 \/ 77 | ___ / 9 \/ 77 || / ____________\ | 3 / - + ------ | ___ \/ 3 *3 / - + ------ || | 3 / - + ------ |\/ 3 *3 / - + ------ ___ || | / ____ | | 1 \/ 2 2 | \/ 3 \/ 2 2 || | 1 \/ 2 2 | \/ 2 2 \/ 3 || | 1 / 9 \/ 77 | |- ------------------- - ----------------- + I*|------------------- - -----------------------||*|- ------------------- - ----------------- + I*|----------------------- - -------------------||*|----------------- + 3 / - + ------ | | ____________ 2 | ____________ 2 || | ____________ 2 | 2 ____________|| | ____________ \/ 2 2 | | / ____ | / ____ || | / ____ | / ____ || | / ____ | | / 9 \/ 77 | / 9 \/ 77 || | / 9 \/ 77 | / 9 \/ 77 || | / 9 \/ 77 | | 2*3 / - + ------ |2*3 / - + ------ || | 2*3 / - + ------ | 2*3 / - + ------ || |3 / - + ------ | \ \/ 2 2 \ \/ 2 2 // \ \/ 2 2 \ \/ 2 2 // \\/ 2 2 /
$$\left(- \frac{\sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}{2} - \frac{1}{2 \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}} + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}{2} - \frac{1}{2 \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}{2} + \frac{\sqrt{3}}{2 \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}\right)\right) \left(\frac{1}{\sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}} + \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}\right)$$
=
9
$$9$$
9
Respuesta rápida
[src]
____________ / ____________\
/ ____ | / ____ |
/ 9 \/ 77 | ___ / 9 \/ 77 |
3 / - + ------ | ___ \/ 3 *3 / - + ------ |
1 \/ 2 2 | \/ 3 \/ 2 2 |
x1 = - ------------------- - ----------------- + I*|------------------- - -----------------------|
____________ 2 | ____________ 2 |
/ ____ | / ____ |
/ 9 \/ 77 | / 9 \/ 77 |
2*3 / - + ------ |2*3 / - + ------ |
\/ 2 2 \ \/ 2 2 /
$$x_{1} = - \frac{\sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}{2} - \frac{1}{2 \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}{2} + \frac{\sqrt{3}}{2 \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}\right)$$
____________ / ____________ \
/ ____ | / ____ |
/ 9 \/ 77 | ___ / 9 \/ 77 |
3 / - + ------ |\/ 3 *3 / - + ------ ___ |
1 \/ 2 2 | \/ 2 2 \/ 3 |
x2 = - ------------------- - ----------------- + I*|----------------------- - -------------------|
____________ 2 | 2 ____________|
/ ____ | / ____ |
/ 9 \/ 77 | / 9 \/ 77 |
2*3 / - + ------ | 2*3 / - + ------ |
\/ 2 2 \ \/ 2 2 /
$$x_{2} = - \frac{\sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}{2} - \frac{1}{2 \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}} + i \left(- \frac{\sqrt{3}}{2 \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}} + \frac{\sqrt{3} \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}}{2}\right)$$
____________
/ ____
1 / 9 \/ 77
x3 = ----------------- + 3 / - + ------
____________ \/ 2 2
/ ____
/ 9 \/ 77
3 / - + ------
\/ 2 2
$$x_{3} = \frac{1}{\sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}} + \sqrt[3]{\frac{\sqrt{77}}{2} + \frac{9}{2}}$$
x3 = (sqrt(77)/2 + 9/2)^(-1/3) + (sqrt(77)/2 + 9/2)^(1/3)
Respuesta numérica
[src]
x1 = 2.55414921860077
x2 = -1.27707460930039 - 1.37577566236622*i
x3 = -1.27707460930039 + 1.37577566236622*i
x3 = -1.27707460930039 + 1.37577566236622*i