Sr Examen
x^3+10*x^2-10=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
$$p = \frac{b}{a}$$
$$p = 10$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = -10$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -10$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = -10$$
$$p x^{2} + q x + v + x^{3} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = 10$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = -10$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -10$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = -10$$
Respuesta rápida
[src]
/ 1 \ / / ______\\ / / 1 \ / / ______\\ / / ______\\\ / / ______\\
100*re|----------------------------------------| | |3*\/ 1119 || | 100*im|----------------------------------------| | |3*\/ 1119 || | |3*\/ 1119 ||| | |3*\/ 1119 ||
| _____________________ / ___\| |atan|----------|| | | _____________________ / ___\| |atan|----------|| |atan|----------||| |atan|----------||
|3 / ______ | 1 I*\/ 3 || | \ 173 /| | |3 / ______ | 1 I*\/ 3 || | \ 173 /| ___ | \ 173 /|| ___ | \ 173 /|
|\/ 865 + 15*I*\/ 1119 *|- - - -------|| 5*cos|----------------| | |\/ 865 + 15*I*\/ 1119 *|- - - -------|| 5*sin|----------------| 5*\/ 3 *cos|----------------|| 5*\/ 3 *sin|----------------|
10 \ \ 2 2 // \ 3 / | \ \ 2 2 // \ 3 / \ 3 /| \ 3 /
x1 = - -- - ------------------------------------------------ + ----------------------- + I*|- ------------------------------------------------ + ----------------------- + -----------------------------| - -----------------------------
3 3 3 \ 3 3 3 / 3
$$x_{1} = - \frac{10}{3} - \frac{5 \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3} - \frac{100 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{865 + 15 \sqrt{1119} i}}\right)}}{3} + \frac{5 \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3} + i \left(- \frac{100 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{865 + 15 \sqrt{1119} i}}\right)}}{3} + \frac{5 \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3} + \frac{5 \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3}\right)$$
/ 1 \ / / ______\\ / / 1 \ / / ______\\ / / ______\\\ / / ______\\
100*re|----------------------------------------| | |3*\/ 1119 || | 100*im|----------------------------------------| | |3*\/ 1119 || | |3*\/ 1119 ||| | |3*\/ 1119 ||
| _____________________ / ___\| |atan|----------|| | | _____________________ / ___\| |atan|----------|| |atan|----------||| |atan|----------||
|3 / ______ | 1 I*\/ 3 || | \ 173 /| | |3 / ______ | 1 I*\/ 3 || | \ 173 /| ___ | \ 173 /|| ___ | \ 173 /|
|\/ 865 + 15*I*\/ 1119 *|- - + -------|| 5*cos|----------------| | |\/ 865 + 15*I*\/ 1119 *|- - + -------|| 5*sin|----------------| 5*\/ 3 *cos|----------------|| 5*\/ 3 *sin|----------------|
10 \ \ 2 2 // \ 3 / | \ \ 2 2 // \ 3 / \ 3 /| \ 3 /
x2 = - -- - ------------------------------------------------ + ----------------------- + I*|- ------------------------------------------------ + ----------------------- - -----------------------------| + -----------------------------
3 3 3 \ 3 3 3 / 3
$$x_{2} = - \frac{10}{3} + \frac{5 \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3} + \frac{5 \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3} - \frac{100 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{865 + 15 \sqrt{1119} i}}\right)}}{3} + i \left(- \frac{5 \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3} + \frac{5 \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3} - \frac{100 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{865 + 15 \sqrt{1119} i}}\right)}}{3}\right)$$
_____________________
3 / ______
10 100 \/ 865 + 15*I*\/ 1119
x3 = - -- - -------------------------- - ------------------------
3 _____________________ 3
3 / ______
3*\/ 865 + 15*I*\/ 1119
$$x_{3} = - \frac{10}{3} - \frac{\sqrt[3]{865 + 15 \sqrt{1119} i}}{3} - \frac{100}{3 \sqrt[3]{865 + 15 \sqrt{1119} i}}$$
x3 = -10/3 - (865 + 15*sqrt(1119)*i)^(1/3)/3 - 100/(3*(865 + 15*sqrt(1119)*i)^(1/3))
Suma y producto de raíces
[src]
suma
/ 1 \ / / ______\\ / / 1 \ / / ______\\ / / ______\\\ / / ______\\ / 1 \ / / ______\\ / / 1 \ / / ______\\ / / ______\\\ / / ______\\
100*re|----------------------------------------| | |3*\/ 1119 || | 100*im|----------------------------------------| | |3*\/ 1119 || | |3*\/ 1119 ||| | |3*\/ 1119 || 100*re|----------------------------------------| | |3*\/ 1119 || | 100*im|----------------------------------------| | |3*\/ 1119 || | |3*\/ 1119 ||| | |3*\/ 1119 ||
| _____________________ / ___\| |atan|----------|| | | _____________________ / ___\| |atan|----------|| |atan|----------||| |atan|----------|| | _____________________ / ___\| |atan|----------|| | | _____________________ / ___\| |atan|----------|| |atan|----------||| |atan|----------||
|3 / ______ | 1 I*\/ 3 || | \ 173 /| | |3 / ______ | 1 I*\/ 3 || | \ 173 /| ___ | \ 173 /|| ___ | \ 173 /| |3 / ______ | 1 I*\/ 3 || | \ 173 /| | |3 / ______ | 1 I*\/ 3 || | \ 173 /| ___ | \ 173 /|| ___ | \ 173 /| _____________________
|\/ 865 + 15*I*\/ 1119 *|- - - -------|| 5*cos|----------------| | |\/ 865 + 15*I*\/ 1119 *|- - - -------|| 5*sin|----------------| 5*\/ 3 *cos|----------------|| 5*\/ 3 *sin|----------------| |\/ 865 + 15*I*\/ 1119 *|- - + -------|| 5*cos|----------------| | |\/ 865 + 15*I*\/ 1119 *|- - + -------|| 5*sin|----------------| 5*\/ 3 *cos|----------------|| 5*\/ 3 *sin|----------------| 3 / ______
10 \ \ 2 2 // \ 3 / | \ \ 2 2 // \ 3 / \ 3 /| \ 3 / 10 \ \ 2 2 // \ 3 / | \ \ 2 2 // \ 3 / \ 3 /| \ 3 / 10 100 \/ 865 + 15*I*\/ 1119
- -- - ------------------------------------------------ + ----------------------- + I*|- ------------------------------------------------ + ----------------------- + -----------------------------| - ----------------------------- + - -- - ------------------------------------------------ + ----------------------- + I*|- ------------------------------------------------ + ----------------------- - -----------------------------| + ----------------------------- + - -- - -------------------------- - ------------------------
3 3 3 \ 3 3 3 / 3 3 3 3 \ 3 3 3 / 3 3 _____________________ 3
3 / ______
3*\/ 865 + 15*I*\/ 1119
$$\left(\left(- \frac{10}{3} - \frac{5 \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3} - \frac{100 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{865 + 15 \sqrt{1119} i}}\right)}}{3} + \frac{5 \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3} + i \left(- \frac{100 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{865 + 15 \sqrt{1119} i}}\right)}}{3} + \frac{5 \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3} + \frac{5 \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3}\right)\right) + \left(- \frac{10}{3} + \frac{5 \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3} + \frac{5 \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3} - \frac{100 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{865 + 15 \sqrt{1119} i}}\right)}}{3} + i \left(- \frac{5 \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3} + \frac{5 \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3} - \frac{100 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{865 + 15 \sqrt{1119} i}}\right)}}{3}\right)\right)\right) + \left(- \frac{10}{3} - \frac{\sqrt[3]{865 + 15 \sqrt{1119} i}}{3} - \frac{100}{3 \sqrt[3]{865 + 15 \sqrt{1119} i}}\right)$$
=
/ 1 \ / 1 \ / / ______\\ / / 1 \ / / ______\\ / / ______\\\ / / 1 \ / / ______\\ / / ______\\\
100*re|----------------------------------------| 100*re|----------------------------------------| | |3*\/ 1119 || | 100*im|----------------------------------------| | |3*\/ 1119 || | |3*\/ 1119 ||| | 100*im|----------------------------------------| | |3*\/ 1119 || | |3*\/ 1119 |||
| _____________________ / ___\| | _____________________ / ___\| |atan|----------|| | | _____________________ / ___\| |atan|----------|| |atan|----------||| | | _____________________ / ___\| |atan|----------|| |atan|----------|||
|3 / ______ | 1 I*\/ 3 || |3 / ______ | 1 I*\/ 3 || _____________________ | \ 173 /| | |3 / ______ | 1 I*\/ 3 || | \ 173 /| ___ | \ 173 /|| | |3 / ______ | 1 I*\/ 3 || | \ 173 /| ___ | \ 173 /||
|\/ 865 + 15*I*\/ 1119 *|- - + -------|| |\/ 865 + 15*I*\/ 1119 *|- - - -------|| 3 / ______ 10*cos|----------------| | |\/ 865 + 15*I*\/ 1119 *|- - + -------|| 5*sin|----------------| 5*\/ 3 *cos|----------------|| | |\/ 865 + 15*I*\/ 1119 *|- - - -------|| 5*sin|----------------| 5*\/ 3 *cos|----------------||
100 \ \ 2 2 // \ \ 2 2 // \/ 865 + 15*I*\/ 1119 \ 3 / | \ \ 2 2 // \ 3 / \ 3 /| | \ \ 2 2 // \ 3 / \ 3 /|
-10 - -------------------------- - ------------------------------------------------ - ------------------------------------------------ - ------------------------ + ------------------------ + I*|- ------------------------------------------------ + ----------------------- - -----------------------------| + I*|- ------------------------------------------------ + ----------------------- + -----------------------------|
_____________________ 3 3 3 3 \ 3 3 3 / \ 3 3 3 /
3 / ______
3*\/ 865 + 15*I*\/ 1119
$$-10 - \frac{100 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{865 + 15 \sqrt{1119} i}}\right)}}{3} - \frac{100 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{865 + 15 \sqrt{1119} i}}\right)}}{3} + \frac{10 \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3} - \frac{\sqrt[3]{865 + 15 \sqrt{1119} i}}{3} + i \left(- \frac{5 \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3} + \frac{5 \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3} - \frac{100 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{865 + 15 \sqrt{1119} i}}\right)}}{3}\right) + i \left(- \frac{100 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{865 + 15 \sqrt{1119} i}}\right)}}{3} + \frac{5 \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3} + \frac{5 \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3}\right) - \frac{100}{3 \sqrt[3]{865 + 15 \sqrt{1119} i}}$$
producto
/ / 1 \ / / ______\\ / / 1 \ / / ______\\ / / ______\\\ / / ______\\\ / / 1 \ / / ______\\ / / 1 \ / / ______\\ / / ______\\\ / / ______\\\
| 100*re|----------------------------------------| | |3*\/ 1119 || | 100*im|----------------------------------------| | |3*\/ 1119 || | |3*\/ 1119 ||| | |3*\/ 1119 ||| | 100*re|----------------------------------------| | |3*\/ 1119 || | 100*im|----------------------------------------| | |3*\/ 1119 || | |3*\/ 1119 ||| | |3*\/ 1119 |||
| | _____________________ / ___\| |atan|----------|| | | _____________________ / ___\| |atan|----------|| |atan|----------||| |atan|----------||| | | _____________________ / ___\| |atan|----------|| | | _____________________ / ___\| |atan|----------|| |atan|----------||| |atan|----------|||
| |3 / ______ | 1 I*\/ 3 || | \ 173 /| | |3 / ______ | 1 I*\/ 3 || | \ 173 /| ___ | \ 173 /|| ___ | \ 173 /|| | |3 / ______ | 1 I*\/ 3 || | \ 173 /| | |3 / ______ | 1 I*\/ 3 || | \ 173 /| ___ | \ 173 /|| ___ | \ 173 /|| / _____________________\
| |\/ 865 + 15*I*\/ 1119 *|- - - -------|| 5*cos|----------------| | |\/ 865 + 15*I*\/ 1119 *|- - - -------|| 5*sin|----------------| 5*\/ 3 *cos|----------------|| 5*\/ 3 *sin|----------------|| | |\/ 865 + 15*I*\/ 1119 *|- - + -------|| 5*cos|----------------| | |\/ 865 + 15*I*\/ 1119 *|- - + -------|| 5*sin|----------------| 5*\/ 3 *cos|----------------|| 5*\/ 3 *sin|----------------|| | 3 / ______ |
| 10 \ \ 2 2 // \ 3 / | \ \ 2 2 // \ 3 / \ 3 /| \ 3 /| | 10 \ \ 2 2 // \ 3 / | \ \ 2 2 // \ 3 / \ 3 /| \ 3 /| | 10 100 \/ 865 + 15*I*\/ 1119 |
|- -- - ------------------------------------------------ + ----------------------- + I*|- ------------------------------------------------ + ----------------------- + -----------------------------| - -----------------------------|*|- -- - ------------------------------------------------ + ----------------------- + I*|- ------------------------------------------------ + ----------------------- - -----------------------------| + -----------------------------|*|- -- - -------------------------- - ------------------------|
\ 3 3 3 \ 3 3 3 / 3 / \ 3 3 3 \ 3 3 3 / 3 / | 3 _____________________ 3 |
| 3 / ______ |
\ 3*\/ 865 + 15*I*\/ 1119 /
$$\left(- \frac{10}{3} - \frac{5 \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3} - \frac{100 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{865 + 15 \sqrt{1119} i}}\right)}}{3} + \frac{5 \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3} + i \left(- \frac{100 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{865 + 15 \sqrt{1119} i}}\right)}}{3} + \frac{5 \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3} + \frac{5 \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3}\right)\right) \left(- \frac{10}{3} + \frac{5 \sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3} + \frac{5 \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3} - \frac{100 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{865 + 15 \sqrt{1119} i}}\right)}}{3} + i \left(- \frac{5 \sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3} + \frac{5 \sin{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}}{3} - \frac{100 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{865 + 15 \sqrt{1119} i}}\right)}}{3}\right)\right) \left(- \frac{10}{3} - \frac{\sqrt[3]{865 + 15 \sqrt{1119} i}}{3} - \frac{100}{3 \sqrt[3]{865 + 15 \sqrt{1119} i}}\right)$$
=
/ / / ______\\ / / ______\\\
| | |3*\/ 1119 || | |3*\/ 1119 |||
/ _____________________ / _____________________\\ | |2*atan|----------|| |atan|----------|||
2/3 | 3 / ______ | 3 / ______ || | | \ 173 /| | \ 173 /||
40*5 *\100 + \/ 865 + 15*I*\/ 1119 *\10 + \/ 865 + 15*I*\/ 1119 //*|- cos|------------------| + cos|----------------||
\ \ 3 / \ 3 //
----------------------------------------------------------------------------------------------------------------------------
____________________
3 / ______
27*\/ 173 + 3*I*\/ 1119
$$\frac{40 \cdot 5^{\frac{2}{3}} \left(100 + \left(10 + \sqrt[3]{865 + 15 \sqrt{1119} i}\right) \sqrt[3]{865 + 15 \sqrt{1119} i}\right) \left(- \cos{\left(\frac{2 \operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)} + \cos{\left(\frac{\operatorname{atan}{\left(\frac{3 \sqrt{1119}}{173} \right)}}{3} \right)}\right)}{27 \sqrt[3]{173 + 3 \sqrt{1119} i}}$$
40*5^(2/3)*(100 + (865 + 15*i*sqrt(1119))^(1/3)*(10 + (865 + 15*i*sqrt(1119))^(1/3)))*(-cos(2*atan(3*sqrt(1119)/173)/3) + cos(atan(3*sqrt(1119)/173)/3))/(27*(173 + 3*i*sqrt(1119))^(1/3))
Respuesta numérica
[src]
x1 = -9.89792684930981
x2 = 0.95540135658765
x3 = -1.05747450727784
x3 = -1.05747450727784