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Expresiones semejantes
- x^3-7x^2-20x-78=0
- x^3+7x^2-20x+78=0
- x^3-7x^2+20x+78=0
x^3-7x^2-20x+78=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
3 2 x - 7*x - 20*x + 78 = 0
$$\left(- 20 x + \left(x^{3} - 7 x^{2}\right)\right) + 78 = 0$$
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -7$$
$$q = \frac{c}{a}$$
$$q = -20$$
$$v = \frac{d}{a}$$
$$v = 78$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 7$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -20$$
$$x_{1} x_{2} x_{3} = 78$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = -7$$
$$q = \frac{c}{a}$$
$$q = -20$$
$$v = \frac{d}{a}$$
$$v = 78$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 7$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -20$$
$$x_{1} x_{2} x_{3} = 78$$
Respuesta rápida
[src]
/ 1 \ / / 1 \ / / _______\\ / / _______\\\ / / _______\\ / / _______\\
109*re|---------------------------------------| | 109*im|---------------------------------------| | |9*\/ 15909 || | |9*\/ 15909 ||| | |9*\/ 15909 || | |9*\/ 15909 ||
| ____________________ / ___\| | | ____________________ / ___\| |atan|-----------|| |atan|-----------||| |atan|-----------|| |atan|-----------||
|3 / _______ | 1 I*\/ 3 || | |3 / _______ | 1 I*\/ 3 || _____ | \ 80 /| _____ | \ 80 /|| _____ | \ 80 /| _____ | \ 80 /|
|\/ 80 + 9*I*\/ 15909 *|- - - -------|| | |\/ 80 + 9*I*\/ 15909 *|- - - -------|| \/ 109 *sin|-----------------| \/ 327 *cos|-----------------|| \/ 327 *sin|-----------------| \/ 109 *cos|-----------------|
7 \ \ 2 2 // | \ \ 2 2 // \ 3 / \ 3 /| \ 3 / \ 3 /
x1 = - - ----------------------------------------------- + I*|- ----------------------------------------------- + ------------------------------ + ------------------------------| - ------------------------------ + ------------------------------
3 3 \ 3 6 6 / 6 6
$$x_{1} = - \frac{\sqrt{327} \sin{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{6} - \frac{109 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{80 + 9 \sqrt{15909} i}}\right)}}{3} + \frac{\sqrt{109} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{6} + \frac{7}{3} + i \left(- \frac{109 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{80 + 9 \sqrt{15909} i}}\right)}}{3} + \frac{\sqrt{109} \sin{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{6} + \frac{\sqrt{327} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{6}\right)$$
/ 1 \ / / 1 \ / / _______\\ / / _______\\\ / / _______\\ / / _______\\
109*re|---------------------------------------| | 109*im|---------------------------------------| | |9*\/ 15909 || | |9*\/ 15909 ||| | |9*\/ 15909 || | |9*\/ 15909 ||
| ____________________ / ___\| | | ____________________ / ___\| |atan|-----------|| |atan|-----------||| |atan|-----------|| |atan|-----------||
|3 / _______ | 1 I*\/ 3 || | |3 / _______ | 1 I*\/ 3 || _____ | \ 80 /| _____ | \ 80 /|| _____ | \ 80 /| _____ | \ 80 /|
|\/ 80 + 9*I*\/ 15909 *|- - + -------|| | |\/ 80 + 9*I*\/ 15909 *|- - + -------|| \/ 327 *cos|-----------------| \/ 109 *sin|-----------------|| \/ 109 *cos|-----------------| \/ 327 *sin|-----------------|
7 \ \ 2 2 // | \ \ 2 2 // \ 3 / \ 3 /| \ 3 / \ 3 /
x2 = - - ----------------------------------------------- + I*|- ----------------------------------------------- - ------------------------------ + ------------------------------| + ------------------------------ + ------------------------------
3 3 \ 3 6 6 / 6 6
$$x_{2} = \frac{\sqrt{327} \sin{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{6} + \frac{\sqrt{109} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{6} + \frac{7}{3} - \frac{109 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{80 + 9 \sqrt{15909} i}}\right)}}{3} + i \left(- \frac{\sqrt{327} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{6} + \frac{\sqrt{109} \sin{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{6} - \frac{109 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{80 + 9 \sqrt{15909} i}}\right)}}{3}\right)$$
____________________
3 / _______
7 109 \/ 80 + 9*I*\/ 15909
x3 = - - ------------------------- - -----------------------
3 ____________________ 3
3 / _______
3*\/ 80 + 9*I*\/ 15909
$$x_{3} = \frac{7}{3} - \frac{\sqrt[3]{80 + 9 \sqrt{15909} i}}{3} - \frac{109}{3 \sqrt[3]{80 + 9 \sqrt{15909} i}}$$
x3 = 7/3 - (80 + 9*sqrt(15909)*i)^(1/3)/3 - 109/(3*(80 + 9*sqrt(15909)*i)^(1/3))
Suma y producto de raíces
[src]
suma
/ 1 \ / / 1 \ / / _______\\ / / _______\\\ / / _______\\ / / _______\\ / 1 \ / / 1 \ / / _______\\ / / _______\\\ / / _______\\ / / _______\\
109*re|---------------------------------------| | 109*im|---------------------------------------| | |9*\/ 15909 || | |9*\/ 15909 ||| | |9*\/ 15909 || | |9*\/ 15909 || 109*re|---------------------------------------| | 109*im|---------------------------------------| | |9*\/ 15909 || | |9*\/ 15909 ||| | |9*\/ 15909 || | |9*\/ 15909 ||
| ____________________ / ___\| | | ____________________ / ___\| |atan|-----------|| |atan|-----------||| |atan|-----------|| |atan|-----------|| | ____________________ / ___\| | | ____________________ / ___\| |atan|-----------|| |atan|-----------||| |atan|-----------|| |atan|-----------||
|3 / _______ | 1 I*\/ 3 || | |3 / _______ | 1 I*\/ 3 || _____ | \ 80 /| _____ | \ 80 /|| _____ | \ 80 /| _____ | \ 80 /| |3 / _______ | 1 I*\/ 3 || | |3 / _______ | 1 I*\/ 3 || _____ | \ 80 /| _____ | \ 80 /|| _____ | \ 80 /| _____ | \ 80 /| ____________________
|\/ 80 + 9*I*\/ 15909 *|- - - -------|| | |\/ 80 + 9*I*\/ 15909 *|- - - -------|| \/ 109 *sin|-----------------| \/ 327 *cos|-----------------|| \/ 327 *sin|-----------------| \/ 109 *cos|-----------------| |\/ 80 + 9*I*\/ 15909 *|- - + -------|| | |\/ 80 + 9*I*\/ 15909 *|- - + -------|| \/ 327 *cos|-----------------| \/ 109 *sin|-----------------|| \/ 109 *cos|-----------------| \/ 327 *sin|-----------------| 3 / _______
7 \ \ 2 2 // | \ \ 2 2 // \ 3 / \ 3 /| \ 3 / \ 3 / 7 \ \ 2 2 // | \ \ 2 2 // \ 3 / \ 3 /| \ 3 / \ 3 / 7 109 \/ 80 + 9*I*\/ 15909
- - ----------------------------------------------- + I*|- ----------------------------------------------- + ------------------------------ + ------------------------------| - ------------------------------ + ------------------------------ + - - ----------------------------------------------- + I*|- ----------------------------------------------- - ------------------------------ + ------------------------------| + ------------------------------ + ------------------------------ + - - ------------------------- - -----------------------
3 3 \ 3 6 6 / 6 6 3 3 \ 3 6 6 / 6 6 3 ____________________ 3
3 / _______
3*\/ 80 + 9*I*\/ 15909
$$\left(\left(- \frac{\sqrt{327} \sin{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{6} - \frac{109 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{80 + 9 \sqrt{15909} i}}\right)}}{3} + \frac{\sqrt{109} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{6} + \frac{7}{3} + i \left(- \frac{109 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{80 + 9 \sqrt{15909} i}}\right)}}{3} + \frac{\sqrt{109} \sin{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{6} + \frac{\sqrt{327} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{6}\right)\right) + \left(\frac{\sqrt{327} \sin{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{6} + \frac{\sqrt{109} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{6} + \frac{7}{3} - \frac{109 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{80 + 9 \sqrt{15909} i}}\right)}}{3} + i \left(- \frac{\sqrt{327} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{6} + \frac{\sqrt{109} \sin{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{6} - \frac{109 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{80 + 9 \sqrt{15909} i}}\right)}}{3}\right)\right)\right) + \left(\frac{7}{3} - \frac{\sqrt[3]{80 + 9 \sqrt{15909} i}}{3} - \frac{109}{3 \sqrt[3]{80 + 9 \sqrt{15909} i}}\right)$$
=
/ 1 \ / 1 \ / / 1 \ / / _______\\ / / _______\\\ / / 1 \ / / _______\\ / / _______\\\ / / _______\\
109*re|---------------------------------------| 109*re|---------------------------------------| | 109*im|---------------------------------------| | |9*\/ 15909 || | |9*\/ 15909 ||| | 109*im|---------------------------------------| | |9*\/ 15909 || | |9*\/ 15909 ||| | |9*\/ 15909 ||
| ____________________ / ___\| | ____________________ / ___\| | | ____________________ / ___\| |atan|-----------|| |atan|-----------||| | | ____________________ / ___\| |atan|-----------|| |atan|-----------||| |atan|-----------||
|3 / _______ | 1 I*\/ 3 || |3 / _______ | 1 I*\/ 3 || ____________________ | |3 / _______ | 1 I*\/ 3 || _____ | \ 80 /| _____ | \ 80 /|| | |3 / _______ | 1 I*\/ 3 || _____ | \ 80 /| _____ | \ 80 /|| _____ | \ 80 /|
|\/ 80 + 9*I*\/ 15909 *|- - + -------|| |\/ 80 + 9*I*\/ 15909 *|- - - -------|| 3 / _______ | |\/ 80 + 9*I*\/ 15909 *|- - + -------|| \/ 327 *cos|-----------------| \/ 109 *sin|-----------------|| | |\/ 80 + 9*I*\/ 15909 *|- - - -------|| \/ 109 *sin|-----------------| \/ 327 *cos|-----------------|| \/ 109 *cos|-----------------|
109 \ \ 2 2 // \ \ 2 2 // \/ 80 + 9*I*\/ 15909 | \ \ 2 2 // \ 3 / \ 3 /| | \ \ 2 2 // \ 3 / \ 3 /| \ 3 /
7 - ------------------------- - ----------------------------------------------- - ----------------------------------------------- - ----------------------- + I*|- ----------------------------------------------- - ------------------------------ + ------------------------------| + I*|- ----------------------------------------------- + ------------------------------ + ------------------------------| + ------------------------------
____________________ 3 3 3 \ 3 6 6 / \ 3 6 6 / 3
3 / _______
3*\/ 80 + 9*I*\/ 15909
$$- \frac{109 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{80 + 9 \sqrt{15909} i}}\right)}}{3} - \frac{109 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{80 + 9 \sqrt{15909} i}}\right)}}{3} + \frac{\sqrt{109} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{3} + 7 - \frac{\sqrt[3]{80 + 9 \sqrt{15909} i}}{3} + i \left(- \frac{\sqrt{327} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{6} + \frac{\sqrt{109} \sin{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{6} - \frac{109 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{80 + 9 \sqrt{15909} i}}\right)}}{3}\right) + i \left(- \frac{109 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{80 + 9 \sqrt{15909} i}}\right)}}{3} + \frac{\sqrt{109} \sin{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{6} + \frac{\sqrt{327} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{6}\right) - \frac{109}{3 \sqrt[3]{80 + 9 \sqrt{15909} i}}$$
producto
/ / 1 \ / / 1 \ / / _______\\ / / _______\\\ / / _______\\ / / _______\\\ / / 1 \ / / 1 \ / / _______\\ / / _______\\\ / / _______\\ / / _______\\\
| 109*re|---------------------------------------| | 109*im|---------------------------------------| | |9*\/ 15909 || | |9*\/ 15909 ||| | |9*\/ 15909 || | |9*\/ 15909 ||| | 109*re|---------------------------------------| | 109*im|---------------------------------------| | |9*\/ 15909 || | |9*\/ 15909 ||| | |9*\/ 15909 || | |9*\/ 15909 |||
| | ____________________ / ___\| | | ____________________ / ___\| |atan|-----------|| |atan|-----------||| |atan|-----------|| |atan|-----------||| | | ____________________ / ___\| | | ____________________ / ___\| |atan|-----------|| |atan|-----------||| |atan|-----------|| |atan|-----------|||
| |3 / _______ | 1 I*\/ 3 || | |3 / _______ | 1 I*\/ 3 || _____ | \ 80 /| _____ | \ 80 /|| _____ | \ 80 /| _____ | \ 80 /|| | |3 / _______ | 1 I*\/ 3 || | |3 / _______ | 1 I*\/ 3 || _____ | \ 80 /| _____ | \ 80 /|| _____ | \ 80 /| _____ | \ 80 /|| / ____________________\
| |\/ 80 + 9*I*\/ 15909 *|- - - -------|| | |\/ 80 + 9*I*\/ 15909 *|- - - -------|| \/ 109 *sin|-----------------| \/ 327 *cos|-----------------|| \/ 327 *sin|-----------------| \/ 109 *cos|-----------------|| | |\/ 80 + 9*I*\/ 15909 *|- - + -------|| | |\/ 80 + 9*I*\/ 15909 *|- - + -------|| \/ 327 *cos|-----------------| \/ 109 *sin|-----------------|| \/ 109 *cos|-----------------| \/ 327 *sin|-----------------|| | 3 / _______ |
|7 \ \ 2 2 // | \ \ 2 2 // \ 3 / \ 3 /| \ 3 / \ 3 /| |7 \ \ 2 2 // | \ \ 2 2 // \ 3 / \ 3 /| \ 3 / \ 3 /| |7 109 \/ 80 + 9*I*\/ 15909 |
|- - ----------------------------------------------- + I*|- ----------------------------------------------- + ------------------------------ + ------------------------------| - ------------------------------ + ------------------------------|*|- - ----------------------------------------------- + I*|- ----------------------------------------------- - ------------------------------ + ------------------------------| + ------------------------------ + ------------------------------|*|- - ------------------------- - -----------------------|
\3 3 \ 3 6 6 / 6 6 / \3 3 \ 3 6 6 / 6 6 / |3 ____________________ 3 |
| 3 / _______ |
\ 3*\/ 80 + 9*I*\/ 15909 /
$$\left(- \frac{\sqrt{327} \sin{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{6} - \frac{109 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{80 + 9 \sqrt{15909} i}}\right)}}{3} + \frac{\sqrt{109} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{6} + \frac{7}{3} + i \left(- \frac{109 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{80 + 9 \sqrt{15909} i}}\right)}}{3} + \frac{\sqrt{109} \sin{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{6} + \frac{\sqrt{327} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{6}\right)\right) \left(\frac{\sqrt{327} \sin{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{6} + \frac{\sqrt{109} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{6} + \frac{7}{3} - \frac{109 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{80 + 9 \sqrt{15909} i}}\right)}}{3} + i \left(- \frac{\sqrt{327} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{6} + \frac{\sqrt{109} \sin{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}}{6} - \frac{109 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{80 + 9 \sqrt{15909} i}}\right)}}{3}\right)\right) \left(\frac{7}{3} - \frac{\sqrt[3]{80 + 9 \sqrt{15909} i}}{3} - \frac{109}{3 \sqrt[3]{80 + 9 \sqrt{15909} i}}\right)$$
=
/ / / _______\\ / / _______\\\
| | |9*\/ 15909 || | |9*\/ 15909 |||
/ ____________________ / ____________________\\ | |2*atan|-----------|| |atan|-----------|||
| 3 / _______ | 3 / _______ || | | \ 80 /| _____ | \ 80 /||
-\109 - \/ 80 + 9*I*\/ 15909 *\7 - \/ 80 + 9*I*\/ 15909 //*|-60 + 218*cos|-------------------| + 14*\/ 109 *cos|-----------------||
\ \ 3 / \ 3 //
----------------------------------------------------------------------------------------------------------------------------------------
____________________
3 / _______
27*\/ 80 + 9*I*\/ 15909
$$- \frac{\left(109 - \left(7 - \sqrt[3]{80 + 9 \sqrt{15909} i}\right) \sqrt[3]{80 + 9 \sqrt{15909} i}\right) \left(-60 + 218 \cos{\left(\frac{2 \operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)} + 14 \sqrt{109} \cos{\left(\frac{\operatorname{atan}{\left(\frac{9 \sqrt{15909}}{80} \right)}}{3} \right)}\right)}{27 \sqrt[3]{80 + 9 \sqrt{15909} i}}$$
-(109 - (80 + 9*i*sqrt(15909))^(1/3)*(7 - (80 + 9*i*sqrt(15909))^(1/3)))*(-60 + 218*cos(2*atan(9*sqrt(15909)/80)/3) + 14*sqrt(109)*cos(atan(9*sqrt(15909)/80)/3))/(27*(80 + 9*i*sqrt(15909))^(1/3))
Respuesta numérica
[src]
x1 = -3.77433212570879
x2 = 8.27778023870892
x3 = 2.49655188699987
x3 = 2.49655188699987