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x^3+3*x^2-1
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- x^3+3*x^2-1
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Expresiones semejantes
- x^3+3*x^2+1=0
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x^3+3*x^2-1=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
$$p = \frac{b}{a}$$
$$p = 3$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = -1$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -3$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = -1$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 3$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = -1$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -3$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = -1$$
Respuesta rápida
[src]
/pi\ / /pi\ ___ /pi\\ ___ /pi\
cos|--| |sin|--| \/ 3 *cos|--|| \/ 3 *sin|--|
\9 / / 1 \ | \9 / / 1 \ \9 /| \9 /
x1 = -1 + ------- - 3*re|--------------------------------------| + I*|------- - 3*im|--------------------------------------| + -------------| - -------------
2 | _________________| | 2 | _________________| 2 | 2
|/ ___\ / ___ | | |/ ___\ / ___ | |
|| 1 I*\/ 3 | / 27 27*I*\/ 3 | | || 1 I*\/ 3 | / 27 27*I*\/ 3 | |
||- - - -------|*3 / -- + ---------- | | ||- - - -------|*3 / -- + ---------- | |
\\ 2 2 / \/ 2 2 / \ \\ 2 2 / \/ 2 2 / /
$$x_{1} = -1 - \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} + i \left(- 3 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right)$$
/pi\ / /pi\ ___ /pi\\ ___ /pi\
cos|--| |sin|--| \/ 3 *cos|--|| \/ 3 *sin|--|
\9 / / 1 \ | \9 / / 1 \ \9 /| \9 /
x2 = -1 + ------- - 3*re|--------------------------------------| + I*|------- - 3*im|--------------------------------------| - -------------| + -------------
2 | _________________| | 2 | _________________| 2 | 2
|/ ___\ / ___ | | |/ ___\ / ___ | |
|| 1 I*\/ 3 | / 27 27*I*\/ 3 | | || 1 I*\/ 3 | / 27 27*I*\/ 3 | |
||- - + -------|*3 / -- + ---------- | | ||- - + -------|*3 / -- + ---------- | |
\\ 2 2 / \/ 2 2 / \ \\ 2 2 / \/ 2 2 / /
$$x_{2} = -1 + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)}\right)$$
_________________
/ ___
/ 27 27*I*\/ 3
3 / -- + ----------
3 \/ 2 2
x3 = -1 - ---------------------- - ----------------------
_________________ 3
/ ___
/ 27 27*I*\/ 3
3 / -- + ----------
\/ 2 2
$$x_{3} = -1 - \frac{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}{3} - \frac{3}{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}$$
x3 = -1 - (27/2 + 27*sqrt(3)*i/2)^(1/3)/3 - 3/(27/2 + 27*sqrt(3)*i/2)^(1/3)
Suma y producto de raíces
[src]
suma
_________________
/ ___
/pi\ / /pi\ ___ /pi\\ ___ /pi\ /pi\ / /pi\ ___ /pi\\ ___ /pi\ / 27 27*I*\/ 3
cos|--| |sin|--| \/ 3 *cos|--|| \/ 3 *sin|--| cos|--| |sin|--| \/ 3 *cos|--|| \/ 3 *sin|--| 3 / -- + ----------
\9 / / 1 \ | \9 / / 1 \ \9 /| \9 / \9 / / 1 \ | \9 / / 1 \ \9 /| \9 / 3 \/ 2 2
-1 + ------- - 3*re|--------------------------------------| + I*|------- - 3*im|--------------------------------------| + -------------| - ------------- + -1 + ------- - 3*re|--------------------------------------| + I*|------- - 3*im|--------------------------------------| - -------------| + ------------- + -1 - ---------------------- - ----------------------
2 | _________________| | 2 | _________________| 2 | 2 2 | _________________| | 2 | _________________| 2 | 2 _________________ 3
|/ ___\ / ___ | | |/ ___\ / ___ | | |/ ___\ / ___ | | |/ ___\ / ___ | | / ___
|| 1 I*\/ 3 | / 27 27*I*\/ 3 | | || 1 I*\/ 3 | / 27 27*I*\/ 3 | | || 1 I*\/ 3 | / 27 27*I*\/ 3 | | || 1 I*\/ 3 | / 27 27*I*\/ 3 | | / 27 27*I*\/ 3
||- - - -------|*3 / -- + ---------- | | ||- - - -------|*3 / -- + ---------- | | ||- - + -------|*3 / -- + ---------- | | ||- - + -------|*3 / -- + ---------- | | 3 / -- + ----------
\\ 2 2 / \/ 2 2 / \ \\ 2 2 / \/ 2 2 / / \\ 2 2 / \/ 2 2 / \ \\ 2 2 / \/ 2 2 / / \/ 2 2
$$\left(\left(-1 + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)}\right)\right) + \left(-1 - \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} + i \left(- 3 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right)\right)\right) + \left(-1 - \frac{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}{3} - \frac{3}{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)$$
=
_________________
/ ___
/ 27 27*I*\/ 3 / /pi\ ___ /pi\\ / /pi\ ___ /pi\\
3 / -- + ---------- |sin|--| \/ 3 *cos|--|| |sin|--| \/ 3 *cos|--||
3 / 1 \ / 1 \ \/ 2 2 | \9 / / 1 \ \9 /| | \9 / / 1 \ \9 /| /pi\
-3 - ---------------------- - 3*re|--------------------------------------| - 3*re|--------------------------------------| - ---------------------- + I*|------- - 3*im|--------------------------------------| - -------------| + I*|------- - 3*im|--------------------------------------| + -------------| + cos|--|
_________________ | _________________| | _________________| 3 | 2 | _________________| 2 | | 2 | _________________| 2 | \9 /
/ ___ |/ ___\ / ___ | |/ ___\ / ___ | | |/ ___\ / ___ | | | |/ ___\ / ___ | |
/ 27 27*I*\/ 3 || 1 I*\/ 3 | / 27 27*I*\/ 3 | || 1 I*\/ 3 | / 27 27*I*\/ 3 | | || 1 I*\/ 3 | / 27 27*I*\/ 3 | | | || 1 I*\/ 3 | / 27 27*I*\/ 3 | |
3 / -- + ---------- ||- - + -------|*3 / -- + ---------- | ||- - - -------|*3 / -- + ---------- | | ||- - + -------|*3 / -- + ---------- | | | ||- - - -------|*3 / -- + ---------- | |
\/ 2 2 \\ 2 2 / \/ 2 2 / \\ 2 2 / \/ 2 2 / \ \\ 2 2 / \/ 2 2 / / \ \\ 2 2 / \/ 2 2 / /
$$-3 - 3 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} - 3 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + \cos{\left(\frac{\pi}{9} \right)} - \frac{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}{3} + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)}\right) + i \left(- 3 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right) - \frac{3}{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}$$
producto
/ _________________\
| / ___ |
/ /pi\ / /pi\ ___ /pi\\ ___ /pi\\ / /pi\ / /pi\ ___ /pi\\ ___ /pi\\ | / 27 27*I*\/ 3 |
| cos|--| |sin|--| \/ 3 *cos|--|| \/ 3 *sin|--|| | cos|--| |sin|--| \/ 3 *cos|--|| \/ 3 *sin|--|| | 3 / -- + ---------- |
| \9 / / 1 \ | \9 / / 1 \ \9 /| \9 /| | \9 / / 1 \ | \9 / / 1 \ \9 /| \9 /| | 3 \/ 2 2 |
|-1 + ------- - 3*re|--------------------------------------| + I*|------- - 3*im|--------------------------------------| + -------------| - -------------|*|-1 + ------- - 3*re|--------------------------------------| + I*|------- - 3*im|--------------------------------------| - -------------| + -------------|*|-1 - ---------------------- - ----------------------|
| 2 | _________________| | 2 | _________________| 2 | 2 | | 2 | _________________| | 2 | _________________| 2 | 2 | | _________________ 3 |
| |/ ___\ / ___ | | |/ ___\ / ___ | | | | |/ ___\ / ___ | | |/ ___\ / ___ | | | | / ___ |
| || 1 I*\/ 3 | / 27 27*I*\/ 3 | | || 1 I*\/ 3 | / 27 27*I*\/ 3 | | | | || 1 I*\/ 3 | / 27 27*I*\/ 3 | | || 1 I*\/ 3 | / 27 27*I*\/ 3 | | | | / 27 27*I*\/ 3 |
| ||- - - -------|*3 / -- + ---------- | | ||- - - -------|*3 / -- + ---------- | | | | ||- - + -------|*3 / -- + ---------- | | ||- - + -------|*3 / -- + ---------- | | | | 3 / -- + ---------- |
\ \\ 2 2 / \/ 2 2 / \ \\ 2 2 / \/ 2 2 / / / \ \\ 2 2 / \/ 2 2 / \ \\ 2 2 / \/ 2 2 / / / \ \/ 2 2 /
$$\left(-1 - \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} + i \left(- 3 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2}\right)\right) \left(-1 + \frac{\sqrt{3} \sin{\left(\frac{\pi}{9} \right)}}{2} + \frac{\cos{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)} + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\pi}{9} \right)}}{2} + \frac{\sin{\left(\frac{\pi}{9} \right)}}{2} - 3 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)}\right)\right) \left(-1 - \frac{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}{3} - \frac{3}{\sqrt[3]{\frac{27}{2} + \frac{27 \sqrt{3} i}{2}}}\right)$$
=
/ _____________ / _____________\\
/ /pi\\ | 3 ___ 3 / ___ | 2/3 3 / ___ || / ___ /pi\ /pi\\
|2 - 4*sin|--||*\2*\/ 2 + \/ 1 + I*\/ 3 *\2 + 2 *\/ 1 + I*\/ 3 //*|-1 + \/ 3 *sin|--| + cos|--||
\ \18// \ \9 / \9 //
-------------------------------------------------------------------------------------------------------
_____________
3 / ___
4*\/ 1 + I*\/ 3
$$\frac{\left(2 - 4 \sin{\left(\frac{\pi}{18} \right)}\right) \left(2 \sqrt[3]{2} + \sqrt[3]{1 + \sqrt{3} i} \left(2 + 2^{\frac{2}{3}} \sqrt[3]{1 + \sqrt{3} i}\right)\right) \left(-1 + \sqrt{3} \sin{\left(\frac{\pi}{9} \right)} + \cos{\left(\frac{\pi}{9} \right)}\right)}{4 \sqrt[3]{1 + \sqrt{3} i}}$$
(2 - 4*sin(pi/18))*(2*2^(1/3) + (1 + i*sqrt(3))^(1/3)*(2 + 2^(2/3)*(1 + i*sqrt(3))^(1/3)))*(-1 + sqrt(3)*sin(pi/9) + cos(pi/9))/(4*(1 + i*sqrt(3))^(1/3))
Respuesta numérica
[src]
x1 = 0.532088886237956
x2 = -2.87938524157182
x3 = -0.652703644666139
x3 = -0.652703644666139
Gráfico