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Expresiones idénticas
- tres *x^ tres - dos *x^ dos - cinco *x+ diez = cero
- 3 multiplicar por x al cubo menos 2 multiplicar por x al cuadrado menos 5 multiplicar por x más 10 es igual a 0
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- 3*x3-2*x2-5*x+10=0
- 3*x³-2*x²-5*x+10=0
- 3*x en el grado 3-2*x en el grado 2-5*x+10=0
- 3x^3-2x^2-5x+10=0
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- 3*x^3-2*x^2-5*x+10=O
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Expresiones semejantes
- 3*x^3+2*x^2-5*x+10=0
- 3*x^3-2*x^2+5*x+10=0
- 3*x^3-2*x^2-5*x-10=0
3*x^3-2*x^2-5*x+10=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
3 2 3*x - 2*x - 5*x + 10 = 0
$$\left(- 5 x + \left(3 x^{3} - 2 x^{2}\right)\right) + 10 = 0$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$\left(- 5 x + \left(3 x^{3} - 2 x^{2}\right)\right) + 10 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - \frac{2 x^{2}}{3} - \frac{5 x}{3} + \frac{10}{3} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - \frac{2}{3}$$
$$q = \frac{c}{a}$$
$$q = - \frac{5}{3}$$
$$v = \frac{d}{a}$$
$$v = \frac{10}{3}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = \frac{2}{3}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = - \frac{5}{3}$$
$$x_{1} x_{2} x_{3} = \frac{10}{3}$$
$$\left(- 5 x + \left(3 x^{3} - 2 x^{2}\right)\right) + 10 = 0$$
de
$$a x^{3} + b x^{2} + c x + d = 0$$
como ecuación cúbica reducida
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - \frac{2 x^{2}}{3} - \frac{5 x}{3} + \frac{10}{3} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - \frac{2}{3}$$
$$q = \frac{c}{a}$$
$$q = - \frac{5}{3}$$
$$v = \frac{d}{a}$$
$$v = \frac{10}{3}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = \frac{2}{3}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = - \frac{5}{3}$$
$$x_{1} x_{2} x_{3} = \frac{10}{3}$$
Respuesta rápida
[src]
_________________ / _________________\
/ 1072 ______ | ___ / 1072 ______ |
3 / ---- + \/ 1415 | ___ \/ 3 *3 / ---- + \/ 1415 |
2 \/ 27 49 | 49*\/ 3 \/ 27 |
x1 = - + --------------------- + ------------------------ + I*|- ------------------------ + ---------------------------|
9 6 _________________ | _________________ 6 |
/ 1072 ______ | / 1072 ______ |
54*3 / ---- + \/ 1415 | 54*3 / ---- + \/ 1415 |
\/ 27 \ \/ 27 /
$$x_{1} = \frac{49}{54 \sqrt[3]{\sqrt{1415} + \frac{1072}{27}}} + \frac{2}{9} + \frac{\sqrt[3]{\sqrt{1415} + \frac{1072}{27}}}{6} + i \left(- \frac{49 \sqrt{3}}{54 \sqrt[3]{\sqrt{1415} + \frac{1072}{27}}} + \frac{\sqrt{3} \sqrt[3]{\sqrt{1415} + \frac{1072}{27}}}{6}\right)$$
_________________ / _________________ \
/ 1072 ______ | ___ / 1072 ______ |
3 / ---- + \/ 1415 | \/ 3 *3 / ---- + \/ 1415 ___ |
2 \/ 27 49 | \/ 27 49*\/ 3 |
x2 = - + --------------------- + ------------------------ + I*|- --------------------------- + ------------------------|
9 6 _________________ | 6 _________________|
/ 1072 ______ | / 1072 ______ |
54*3 / ---- + \/ 1415 | 54*3 / ---- + \/ 1415 |
\/ 27 \ \/ 27 /
$$x_{2} = \frac{49}{54 \sqrt[3]{\sqrt{1415} + \frac{1072}{27}}} + \frac{2}{9} + \frac{\sqrt[3]{\sqrt{1415} + \frac{1072}{27}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\sqrt{1415} + \frac{1072}{27}}}{6} + \frac{49 \sqrt{3}}{54 \sqrt[3]{\sqrt{1415} + \frac{1072}{27}}}\right)$$
_________________
/ 1072 ______
3 / ---- + \/ 1415
2 49 \/ 27
x3 = - - ------------------------ - ---------------------
9 _________________ 3
/ 1072 ______
27*3 / ---- + \/ 1415
\/ 27
$$x_{3} = - \frac{\sqrt[3]{\sqrt{1415} + \frac{1072}{27}}}{3} - \frac{49}{27 \sqrt[3]{\sqrt{1415} + \frac{1072}{27}}} + \frac{2}{9}$$
x3 = -(sqrt(1415) + 1072/27)^(1/3)/3 - 49/(27*(sqrt(1415) + 1072/27)^(1/3)) + 2/9
Suma y producto de raíces
[src]
suma
_________________ / _________________\ _________________ / _________________ \ _________________
/ 1072 ______ | ___ / 1072 ______ | / 1072 ______ | ___ / 1072 ______ | / 1072 ______
3 / ---- + \/ 1415 | ___ \/ 3 *3 / ---- + \/ 1415 | 3 / ---- + \/ 1415 | \/ 3 *3 / ---- + \/ 1415 ___ | 3 / ---- + \/ 1415
2 \/ 27 49 | 49*\/ 3 \/ 27 | 2 \/ 27 49 | \/ 27 49*\/ 3 | 2 49 \/ 27
- + --------------------- + ------------------------ + I*|- ------------------------ + ---------------------------| + - + --------------------- + ------------------------ + I*|- --------------------------- + ------------------------| + - - ------------------------ - ---------------------
9 6 _________________ | _________________ 6 | 9 6 _________________ | 6 _________________| 9 _________________ 3
/ 1072 ______ | / 1072 ______ | / 1072 ______ | / 1072 ______ | / 1072 ______
54*3 / ---- + \/ 1415 | 54*3 / ---- + \/ 1415 | 54*3 / ---- + \/ 1415 | 54*3 / ---- + \/ 1415 | 27*3 / ---- + \/ 1415
\/ 27 \ \/ 27 / \/ 27 \ \/ 27 / \/ 27
$$\left(- \frac{\sqrt[3]{\sqrt{1415} + \frac{1072}{27}}}{3} - \frac{49}{27 \sqrt[3]{\sqrt{1415} + \frac{1072}{27}}} + \frac{2}{9}\right) + \left(\left(\frac{49}{54 \sqrt[3]{\sqrt{1415} + \frac{1072}{27}}} + \frac{2}{9} + \frac{\sqrt[3]{\sqrt{1415} + \frac{1072}{27}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\sqrt{1415} + \frac{1072}{27}}}{6} + \frac{49 \sqrt{3}}{54 \sqrt[3]{\sqrt{1415} + \frac{1072}{27}}}\right)\right) + \left(\frac{49}{54 \sqrt[3]{\sqrt{1415} + \frac{1072}{27}}} + \frac{2}{9} + \frac{\sqrt[3]{\sqrt{1415} + \frac{1072}{27}}}{6} + i \left(- \frac{49 \sqrt{3}}{54 \sqrt[3]{\sqrt{1415} + \frac{1072}{27}}} + \frac{\sqrt{3} \sqrt[3]{\sqrt{1415} + \frac{1072}{27}}}{6}\right)\right)\right)$$
=
/ _________________\ / _________________ \
| ___ / 1072 ______ | | ___ / 1072 ______ |
| ___ \/ 3 *3 / ---- + \/ 1415 | | \/ 3 *3 / ---- + \/ 1415 ___ |
2 | 49*\/ 3 \/ 27 | | \/ 27 49*\/ 3 |
- + I*|- ------------------------ + ---------------------------| + I*|- --------------------------- + ------------------------|
3 | _________________ 6 | | 6 _________________|
| / 1072 ______ | | / 1072 ______ |
| 54*3 / ---- + \/ 1415 | | 54*3 / ---- + \/ 1415 |
\ \/ 27 / \ \/ 27 /
$$\frac{2}{3} + i \left(- \frac{\sqrt{3} \sqrt[3]{\sqrt{1415} + \frac{1072}{27}}}{6} + \frac{49 \sqrt{3}}{54 \sqrt[3]{\sqrt{1415} + \frac{1072}{27}}}\right) + i \left(- \frac{49 \sqrt{3}}{54 \sqrt[3]{\sqrt{1415} + \frac{1072}{27}}} + \frac{\sqrt{3} \sqrt[3]{\sqrt{1415} + \frac{1072}{27}}}{6}\right)$$
producto
/ _________________ / _________________\\ / _________________ / _________________ \\ / _________________\ | / 1072 ______ | ___ / 1072 ______ || | / 1072 ______ | ___ / 1072 ______ || | / 1072 ______ | | 3 / ---- + \/ 1415 | ___ \/ 3 *3 / ---- + \/ 1415 || | 3 / ---- + \/ 1415 | \/ 3 *3 / ---- + \/ 1415 ___ || | 3 / ---- + \/ 1415 | |2 \/ 27 49 | 49*\/ 3 \/ 27 || |2 \/ 27 49 | \/ 27 49*\/ 3 || |2 49 \/ 27 | |- + --------------------- + ------------------------ + I*|- ------------------------ + ---------------------------||*|- + --------------------- + ------------------------ + I*|- --------------------------- + ------------------------||*|- - ------------------------ - ---------------------| |9 6 _________________ | _________________ 6 || |9 6 _________________ | 6 _________________|| |9 _________________ 3 | | / 1072 ______ | / 1072 ______ || | / 1072 ______ | / 1072 ______ || | / 1072 ______ | | 54*3 / ---- + \/ 1415 | 54*3 / ---- + \/ 1415 || | 54*3 / ---- + \/ 1415 | 54*3 / ---- + \/ 1415 || | 27*3 / ---- + \/ 1415 | \ \/ 27 \ \/ 27 // \ \/ 27 \ \/ 27 // \ \/ 27 /
$$\left(\frac{49}{54 \sqrt[3]{\sqrt{1415} + \frac{1072}{27}}} + \frac{2}{9} + \frac{\sqrt[3]{\sqrt{1415} + \frac{1072}{27}}}{6} + i \left(- \frac{49 \sqrt{3}}{54 \sqrt[3]{\sqrt{1415} + \frac{1072}{27}}} + \frac{\sqrt{3} \sqrt[3]{\sqrt{1415} + \frac{1072}{27}}}{6}\right)\right) \left(\frac{49}{54 \sqrt[3]{\sqrt{1415} + \frac{1072}{27}}} + \frac{2}{9} + \frac{\sqrt[3]{\sqrt{1415} + \frac{1072}{27}}}{6} + i \left(- \frac{\sqrt{3} \sqrt[3]{\sqrt{1415} + \frac{1072}{27}}}{6} + \frac{49 \sqrt{3}}{54 \sqrt[3]{\sqrt{1415} + \frac{1072}{27}}}\right)\right) \left(- \frac{\sqrt[3]{\sqrt{1415} + \frac{1072}{27}}}{3} - \frac{49}{27 \sqrt[3]{\sqrt{1415} + \frac{1072}{27}}} + \frac{2}{9}\right)$$
=
-10/3
$$- \frac{10}{3}$$
-10/3
Respuesta numérica
[src]
x1 = 1.145253113016 + 0.860896777357091*i
x2 = 1.145253113016 - 0.860896777357091*i
x3 = -1.62383955936532
x3 = -1.62383955936532