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Expresiones idénticas
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- x^3+x^2+x-1=O
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Expresiones semejantes
- x^3+x^2-x-1=0
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- x^3-x^2+x-1=0
x^3+x^2+x-1=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
$$p = \frac{b}{a}$$
$$p = 1$$
$$q = \frac{c}{a}$$
$$q = 1$$
$$v = \frac{d}{a}$$
$$v = -1$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -1$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 1$$
$$x_{1} x_{2} x_{3} = -1$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 1$$
$$q = \frac{c}{a}$$
$$q = 1$$
$$v = \frac{d}{a}$$
$$v = -1$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -1$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 1$$
$$x_{1} x_{2} x_{3} = -1$$
Suma y producto de raíces
[src]
suma
_____________ / _____________ \ _____________ / _____________ \
/ ____ | / ____ | / ____ | / ____ |
/ 17 \/ 33 | ___ / 17 \/ 33 | / 17 \/ 33 | ___ / 17 \/ 33 | _____________
3 / -- + ------ | \/ 3 *3 / -- + ------ ___ | 3 / -- + ------ |\/ 3 *3 / -- + ------ ___ | / ____
1 \/ 27 9 1 | \/ 27 9 \/ 3 | 1 \/ 27 9 1 | \/ 27 9 \/ 3 | 1 / 17 \/ 33 2
- - - ------------------ + -------------------- + I*|- ------------------------ - --------------------| + - - - ------------------ + -------------------- + I*|------------------------ + --------------------| + - - + 3 / -- + ------ - --------------------
3 2 _____________ | 2 _____________| 3 2 _____________ | 2 _____________| 3 \/ 27 9 _____________
/ ____ | / ____ | / ____ | / ____ | / ____
/ 17 \/ 33 | / 17 \/ 33 | / 17 \/ 33 | / 17 \/ 33 | / 17 \/ 33
9*3 / -- + ------ | 9*3 / -- + ------ | 9*3 / -- + ------ | 9*3 / -- + ------ | 9*3 / -- + ------
\/ 27 9 \ \/ 27 9 / \/ 27 9 \ \/ 27 9 / \/ 27 9
$$\left(- \frac{1}{3} - \frac{2}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}} + \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}\right) + \left(\left(- \frac{\sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}{2} - \frac{1}{3} + \frac{1}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}{2} - \frac{\sqrt{3}}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}\right)\right) + \left(- \frac{\sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}{2} - \frac{1}{3} + \frac{1}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}} + i \left(\frac{\sqrt{3}}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}} + \frac{\sqrt{3} \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}{2}\right)\right)\right)$$
=
/ _____________ \ / _____________ \
| / ____ | | / ____ |
| ___ / 17 \/ 33 | | ___ / 17 \/ 33 |
|\/ 3 *3 / -- + ------ ___ | | \/ 3 *3 / -- + ------ ___ |
| \/ 27 9 \/ 3 | | \/ 27 9 \/ 3 |
-1 + I*|------------------------ + --------------------| + I*|- ------------------------ - --------------------|
| 2 _____________| | 2 _____________|
| / ____ | | / ____ |
| / 17 \/ 33 | | / 17 \/ 33 |
| 9*3 / -- + ------ | | 9*3 / -- + ------ |
\ \/ 27 9 / \ \/ 27 9 /
$$-1 + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}{2} - \frac{\sqrt{3}}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}\right) + i \left(\frac{\sqrt{3}}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}} + \frac{\sqrt{3} \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}{2}\right)$$
producto
/ _____________ / _____________ \\ / _____________ / _____________ \\ | / ____ | / ____ || | / ____ | / ____ || | / 17 \/ 33 | ___ / 17 \/ 33 || | / 17 \/ 33 | ___ / 17 \/ 33 || / _____________ \ | 3 / -- + ------ | \/ 3 *3 / -- + ------ ___ || | 3 / -- + ------ |\/ 3 *3 / -- + ------ ___ || | / ____ | | 1 \/ 27 9 1 | \/ 27 9 \/ 3 || | 1 \/ 27 9 1 | \/ 27 9 \/ 3 || | 1 / 17 \/ 33 2 | |- - - ------------------ + -------------------- + I*|- ------------------------ - --------------------||*|- - - ------------------ + -------------------- + I*|------------------------ + --------------------||*|- - + 3 / -- + ------ - --------------------| | 3 2 _____________ | 2 _____________|| | 3 2 _____________ | 2 _____________|| | 3 \/ 27 9 _____________| | / ____ | / ____ || | / ____ | / ____ || | / ____ | | / 17 \/ 33 | / 17 \/ 33 || | / 17 \/ 33 | / 17 \/ 33 || | / 17 \/ 33 | | 9*3 / -- + ------ | 9*3 / -- + ------ || | 9*3 / -- + ------ | 9*3 / -- + ------ || | 9*3 / -- + ------ | \ \/ 27 9 \ \/ 27 9 // \ \/ 27 9 \ \/ 27 9 // \ \/ 27 9 /
$$\left(- \frac{\sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}{2} - \frac{1}{3} + \frac{1}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}} + i \left(\frac{\sqrt{3}}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}} + \frac{\sqrt{3} \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}{2}\right)\right) \left(- \frac{\sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}{2} - \frac{1}{3} + \frac{1}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}{2} - \frac{\sqrt{3}}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}\right)\right) \left(- \frac{1}{3} - \frac{2}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}} + \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}\right)$$
=
1
$$1$$
1
Respuesta rápida
[src]
_____________ / _____________ \
/ ____ | / ____ |
/ 17 \/ 33 | ___ / 17 \/ 33 |
3 / -- + ------ | \/ 3 *3 / -- + ------ ___ |
1 \/ 27 9 1 | \/ 27 9 \/ 3 |
x1 = - - - ------------------ + -------------------- + I*|- ------------------------ - --------------------|
3 2 _____________ | 2 _____________|
/ ____ | / ____ |
/ 17 \/ 33 | / 17 \/ 33 |
9*3 / -- + ------ | 9*3 / -- + ------ |
\/ 27 9 \ \/ 27 9 /
$$x_{1} = - \frac{\sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}{2} - \frac{1}{3} + \frac{1}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}} + i \left(- \frac{\sqrt{3} \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}{2} - \frac{\sqrt{3}}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}\right)$$
_____________ / _____________ \
/ ____ | / ____ |
/ 17 \/ 33 | ___ / 17 \/ 33 |
3 / -- + ------ |\/ 3 *3 / -- + ------ ___ |
1 \/ 27 9 1 | \/ 27 9 \/ 3 |
x2 = - - - ------------------ + -------------------- + I*|------------------------ + --------------------|
3 2 _____________ | 2 _____________|
/ ____ | / ____ |
/ 17 \/ 33 | / 17 \/ 33 |
9*3 / -- + ------ | 9*3 / -- + ------ |
\/ 27 9 \ \/ 27 9 /
$$x_{2} = - \frac{\sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}{2} - \frac{1}{3} + \frac{1}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}} + i \left(\frac{\sqrt{3}}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}} + \frac{\sqrt{3} \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}}{2}\right)$$
_____________
/ ____
1 / 17 \/ 33 2
x3 = - - + 3 / -- + ------ - --------------------
3 \/ 27 9 _____________
/ ____
/ 17 \/ 33
9*3 / -- + ------
\/ 27 9
$$x_{3} = - \frac{1}{3} - \frac{2}{9 \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}} + \sqrt[3]{\frac{17}{27} + \frac{\sqrt{33}}{9}}$$
x3 = -1/3 - 2/(9*(17/27 + sqrt(33)/9)^(1/3)) + (17/27 + sqrt(33)/9)^(1/3)
Respuesta numérica
[src]
x1 = -0.771844506346038 + 1.11514250803994*i
x2 = -0.771844506346038 - 1.11514250803994*i
x3 = 0.543689012692076
x3 = 0.543689012692076
Gráfico