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Expresiones semejantes
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x^3+7*x^2-20*x-72=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
3 2 x + 7*x - 20*x - 72 = 0
$$\left(- 20 x + \left(x^{3} + 7 x^{2}\right)\right) - 72 = 0$$
Teorema de Cardano-Vieta
es ecuación cúbica reducida
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 7$$
$$q = \frac{c}{a}$$
$$q = -20$$
$$v = \frac{d}{a}$$
$$v = -72$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -7$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -20$$
$$x_{1} x_{2} x_{3} = -72$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 7$$
$$q = \frac{c}{a}$$
$$q = -20$$
$$v = \frac{d}{a}$$
$$v = -72$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -7$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -20$$
$$x_{1} x_{2} x_{3} = -72$$
Respuesta rápida
[src]
/ 1 \ / / 1 \ \
109*re|--------------------------------------| | 109*im|--------------------------------------| |
| ___________________ / ___\| | | ___________________ / ___\| / / ______\\ / / ______\\| / / ______\\ / / ______\\
|3 / ______ | 1 I*\/ 3 || | |3 / ______ | 1 I*\/ 3 || _____ |atan\18*\/ 3997 /| _____ |atan\18*\/ 3997 /|| _____ |atan\18*\/ 3997 /| _____ |atan\18*\/ 3997 /|
|\/ 1 + 18*I*\/ 3997 *|- - - -------|| | |\/ 1 + 18*I*\/ 3997 *|- - - -------|| \/ 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
$$x_{1} = - \frac{7}{3} - \frac{\sqrt{327} \sin{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{6} - \frac{109 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1 + 18 \sqrt{3997} i}}\right)}}{3} + \frac{\sqrt{109} \cos{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{6} + i \left(- \frac{109 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1 + 18 \sqrt{3997} i}}\right)}}{3} + \frac{\sqrt{109} \sin{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{6} + \frac{\sqrt{327} \cos{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{6}\right)$$
/ 1 \ / / 1 \ \
109*re|--------------------------------------| | 109*im|--------------------------------------| |
| ___________________ / ___\| | | ___________________ / ___\| / / ______\\ / / ______\\| / / ______\\ / / ______\\
|3 / ______ | 1 I*\/ 3 || | |3 / ______ | 1 I*\/ 3 || _____ |atan\18*\/ 3997 /| _____ |atan\18*\/ 3997 /|| _____ |atan\18*\/ 3997 /| _____ |atan\18*\/ 3997 /|
|\/ 1 + 18*I*\/ 3997 *|- - + -------|| | |\/ 1 + 18*I*\/ 3997 *|- - + -------|| \/ 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
$$x_{2} = - \frac{7}{3} + \frac{\sqrt{327} \sin{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{6} + \frac{\sqrt{109} \cos{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{6} - \frac{109 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1 + 18 \sqrt{3997} i}}\right)}}{3} + i \left(- \frac{\sqrt{327} \cos{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{6} + \frac{\sqrt{109} \sin{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{6} - \frac{109 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1 + 18 \sqrt{3997} i}}\right)}}{3}\right)$$
___________________
3 / ______
7 109 \/ 1 + 18*I*\/ 3997
x3 = - - - ------------------------ - ----------------------
3 ___________________ 3
3 / ______
3*\/ 1 + 18*I*\/ 3997
$$x_{3} = - \frac{7}{3} - \frac{\sqrt[3]{1 + 18 \sqrt{3997} i}}{3} - \frac{109}{3 \sqrt[3]{1 + 18 \sqrt{3997} i}}$$
x3 = -7/3 - (1 + 18*sqrt(3997)*i)^(1/3)/3 - 109/(3*(1 + 18*sqrt(3997)*i)^(1/3))
Suma y producto de raíces
[src]
suma
/ 1 \ / / 1 \ \ / 1 \ / / 1 \ \
109*re|--------------------------------------| | 109*im|--------------------------------------| | 109*re|--------------------------------------| | 109*im|--------------------------------------| |
| ___________________ / ___\| | | ___________________ / ___\| / / ______\\ / / ______\\| / / ______\\ / / ______\\ | ___________________ / ___\| | | ___________________ / ___\| / / ______\\ / / ______\\| / / ______\\ / / ______\\
|3 / ______ | 1 I*\/ 3 || | |3 / ______ | 1 I*\/ 3 || _____ |atan\18*\/ 3997 /| _____ |atan\18*\/ 3997 /|| _____ |atan\18*\/ 3997 /| _____ |atan\18*\/ 3997 /| |3 / ______ | 1 I*\/ 3 || | |3 / ______ | 1 I*\/ 3 || _____ |atan\18*\/ 3997 /| _____ |atan\18*\/ 3997 /|| _____ |atan\18*\/ 3997 /| _____ |atan\18*\/ 3997 /| ___________________
|\/ 1 + 18*I*\/ 3997 *|- - - -------|| | |\/ 1 + 18*I*\/ 3997 *|- - - -------|| \/ 109 *sin|-----------------| \/ 327 *cos|-----------------|| \/ 327 *sin|-----------------| \/ 109 *cos|-----------------| |\/ 1 + 18*I*\/ 3997 *|- - + -------|| | |\/ 1 + 18*I*\/ 3997 *|- - + -------|| \/ 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 \/ 1 + 18*I*\/ 3997
- - - ---------------------------------------------- + I*|- ---------------------------------------------- + ------------------------------ + ------------------------------| - ------------------------------ + ------------------------------ + - - - ---------------------------------------------- + I*|- ---------------------------------------------- - ------------------------------ + ------------------------------| + ------------------------------ + ------------------------------ + - - - ------------------------ - ----------------------
3 3 \ 3 6 6 / 6 6 3 3 \ 3 6 6 / 6 6 3 ___________________ 3
3 / ______
3*\/ 1 + 18*I*\/ 3997
$$\left(- \frac{7}{3} - \frac{\sqrt[3]{1 + 18 \sqrt{3997} i}}{3} - \frac{109}{3 \sqrt[3]{1 + 18 \sqrt{3997} i}}\right) + \left(\left(- \frac{7}{3} - \frac{\sqrt{327} \sin{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{6} - \frac{109 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1 + 18 \sqrt{3997} i}}\right)}}{3} + \frac{\sqrt{109} \cos{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{6} + i \left(- \frac{109 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1 + 18 \sqrt{3997} i}}\right)}}{3} + \frac{\sqrt{109} \sin{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{6} + \frac{\sqrt{327} \cos{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{6}\right)\right) + \left(- \frac{7}{3} + \frac{\sqrt{327} \sin{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{6} + \frac{\sqrt{109} \cos{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{6} - \frac{109 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1 + 18 \sqrt{3997} i}}\right)}}{3} + i \left(- \frac{\sqrt{327} \cos{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{6} + \frac{\sqrt{109} \sin{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{6} - \frac{109 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1 + 18 \sqrt{3997} i}}\right)}}{3}\right)\right)\right)$$
=
/ 1 \ / 1 \ / / 1 \ \ / / 1 \ \
109*re|--------------------------------------| 109*re|--------------------------------------| | 109*im|--------------------------------------| | | 109*im|--------------------------------------| |
| ___________________ / ___\| | ___________________ / ___\| | | ___________________ / ___\| / / ______\\ / / ______\\| | | ___________________ / ___\| / / ______\\ / / ______\\| / / ______\\
|3 / ______ | 1 I*\/ 3 || |3 / ______ | 1 I*\/ 3 || ___________________ | |3 / ______ | 1 I*\/ 3 || _____ |atan\18*\/ 3997 /| _____ |atan\18*\/ 3997 /|| | |3 / ______ | 1 I*\/ 3 || _____ |atan\18*\/ 3997 /| _____ |atan\18*\/ 3997 /|| _____ |atan\18*\/ 3997 /|
|\/ 1 + 18*I*\/ 3997 *|- - + -------|| |\/ 1 + 18*I*\/ 3997 *|- - - -------|| 3 / ______ | |\/ 1 + 18*I*\/ 3997 *|- - + -------|| \/ 327 *cos|-----------------| \/ 109 *sin|-----------------|| | |\/ 1 + 18*I*\/ 3997 *|- - - -------|| \/ 109 *sin|-----------------| \/ 327 *cos|-----------------|| \/ 109 *cos|-----------------|
109 \ \ 2 2 // \ \ 2 2 // \/ 1 + 18*I*\/ 3997 | \ \ 2 2 // \ 3 / \ 3 /| | \ \ 2 2 // \ 3 / \ 3 /| \ 3 /
-7 - ------------------------ - ---------------------------------------------- - ---------------------------------------------- - ---------------------- + I*|- ---------------------------------------------- - ------------------------------ + ------------------------------| + I*|- ---------------------------------------------- + ------------------------------ + ------------------------------| + ------------------------------
___________________ 3 3 3 \ 3 6 6 / \ 3 6 6 / 3
3 / ______
3*\/ 1 + 18*I*\/ 3997
$$-7 - \frac{109 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1 + 18 \sqrt{3997} i}}\right)}}{3} - \frac{109 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1 + 18 \sqrt{3997} i}}\right)}}{3} + \frac{\sqrt{109} \cos{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{3} - \frac{\sqrt[3]{1 + 18 \sqrt{3997} i}}{3} + i \left(- \frac{109 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1 + 18 \sqrt{3997} i}}\right)}}{3} + \frac{\sqrt{109} \sin{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{6} + \frac{\sqrt{327} \cos{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{6}\right) + i \left(- \frac{\sqrt{327} \cos{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{6} + \frac{\sqrt{109} \sin{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{6} - \frac{109 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1 + 18 \sqrt{3997} i}}\right)}}{3}\right) - \frac{109}{3 \sqrt[3]{1 + 18 \sqrt{3997} i}}$$
producto
/ / 1 \ / / 1 \ \ \ / / 1 \ / / 1 \ \ \
| 109*re|--------------------------------------| | 109*im|--------------------------------------| | | | 109*re|--------------------------------------| | 109*im|--------------------------------------| | |
| | ___________________ / ___\| | | ___________________ / ___\| / / ______\\ / / ______\\| / / ______\\ / / ______\\| | | ___________________ / ___\| | | ___________________ / ___\| / / ______\\ / / ______\\| / / ______\\ / / ______\\|
| |3 / ______ | 1 I*\/ 3 || | |3 / ______ | 1 I*\/ 3 || _____ |atan\18*\/ 3997 /| _____ |atan\18*\/ 3997 /|| _____ |atan\18*\/ 3997 /| _____ |atan\18*\/ 3997 /|| | |3 / ______ | 1 I*\/ 3 || | |3 / ______ | 1 I*\/ 3 || _____ |atan\18*\/ 3997 /| _____ |atan\18*\/ 3997 /|| _____ |atan\18*\/ 3997 /| _____ |atan\18*\/ 3997 /|| / ___________________\
| |\/ 1 + 18*I*\/ 3997 *|- - - -------|| | |\/ 1 + 18*I*\/ 3997 *|- - - -------|| \/ 109 *sin|-----------------| \/ 327 *cos|-----------------|| \/ 327 *sin|-----------------| \/ 109 *cos|-----------------|| | |\/ 1 + 18*I*\/ 3997 *|- - + -------|| | |\/ 1 + 18*I*\/ 3997 *|- - + -------|| \/ 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 \/ 1 + 18*I*\/ 3997 |
|- - - ---------------------------------------------- + I*|- ---------------------------------------------- + ------------------------------ + ------------------------------| - ------------------------------ + ------------------------------|*|- - - ---------------------------------------------- + I*|- ---------------------------------------------- - ------------------------------ + ------------------------------| + ------------------------------ + ------------------------------|*|- - - ------------------------ - ----------------------|
\ 3 3 \ 3 6 6 / 6 6 / \ 3 3 \ 3 6 6 / 6 6 / | 3 ___________________ 3 |
| 3 / ______ |
\ 3*\/ 1 + 18*I*\/ 3997 /
$$\left(- \frac{7}{3} - \frac{\sqrt{327} \sin{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{6} - \frac{109 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1 + 18 \sqrt{3997} i}}\right)}}{3} + \frac{\sqrt{109} \cos{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{6} + i \left(- \frac{109 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1 + 18 \sqrt{3997} i}}\right)}}{3} + \frac{\sqrt{109} \sin{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{6} + \frac{\sqrt{327} \cos{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{6}\right)\right) \left(- \frac{7}{3} + \frac{\sqrt{327} \sin{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{6} + \frac{\sqrt{109} \cos{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{6} - \frac{109 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1 + 18 \sqrt{3997} i}}\right)}}{3} + i \left(- \frac{\sqrt{327} \cos{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{6} + \frac{\sqrt{109} \sin{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}}{6} - \frac{109 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{1 + 18 \sqrt{3997} i}}\right)}}{3}\right)\right) \left(- \frac{7}{3} - \frac{\sqrt[3]{1 + 18 \sqrt{3997} i}}{3} - \frac{109}{3 \sqrt[3]{1 + 18 \sqrt{3997} i}}\right)$$
=
/ ___________________ / ___________________\\ / / / ______\\ / / ______\\\
| 3 / ______ | 3 / ______ || | |2*atan\18*\/ 3997 /| _____ |atan\18*\/ 3997 /||
\109 + \/ 1 + 18*I*\/ 3997 *\7 + \/ 1 + 18*I*\/ 3997 //*|60 - 218*cos|-------------------| + 14*\/ 109 *cos|-----------------||
\ \ 3 / \ 3 //
-----------------------------------------------------------------------------------------------------------------------------------
___________________
3 / ______
27*\/ 1 + 18*I*\/ 3997
$$\frac{\left(109 + \sqrt[3]{1 + 18 \sqrt{3997} i} \left(7 + \sqrt[3]{1 + 18 \sqrt{3997} i}\right)\right) \left(- 218 \cos{\left(\frac{2 \operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)} + 60 + 14 \sqrt{109} \cos{\left(\frac{\operatorname{atan}{\left(18 \sqrt{3997} \right)}}{3} \right)}\right)}{27 \sqrt[3]{1 + 18 \sqrt{3997} i}}$$
(109 + (1 + 18*i*sqrt(3997))^(1/3)*(7 + (1 + 18*i*sqrt(3997))^(1/3)))*(60 - 218*cos(2*atan(18*sqrt(3997))/3) + 14*sqrt(109)*cos(atan(18*sqrt(3997))/3))/(27*(1 + 18*i*sqrt(3997))^(1/3))
Respuesta numérica
[src]
x1 = 3.69336081331646
x2 = -8.36206621620005
x3 = -2.33129459711642
x3 = -2.33129459711642