Произведение корней x^3-2022*x+sqrt(2021)=0
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Решение
Сумма и произведение корней
[src]
сумма
/ / / ______\\ / / ______\\\ / / ______\\ / / ______\\ / / / ______\\ / / ______\\\ / / ______\\ / / ______\\
| | |20205*\/ 6063 || | |20205*\/ 6063 ||| | |20205*\/ 6063 || | |20205*\/ 6063 || | | |20205*\/ 6063 || | |20205*\/ 6063 ||| | |20205*\/ 6063 || | |20205*\/ 6063 || ______________________________
| |atan|--------------|| |atan|--------------||| |atan|--------------|| |atan|--------------|| | |atan|--------------|| |atan|--------------||| |atan|--------------|| |atan|--------------|| / ______ ___
| _____ | \ 2021 /| ______ | \ 2021 /|| _____ | \ 2021 /| ______ | \ 2021 /| | _____ | \ 2021 /| ______ | \ 2021 /|| _____ | \ 2021 /| ______ | \ 2021 /| / 27*\/ 2021 545535*I*\/ 3
| \/ 674 *sin|--------------------| \/ 2022 *cos|--------------------|| \/ 674 *cos|--------------------| \/ 2022 *sin|--------------------| | \/ 674 *sin|--------------------| \/ 2022 *cos|--------------------|| \/ 674 *cos|--------------------| \/ 2022 *sin|--------------------| 3 / ----------- + --------------
/ 1 \ | / 1 \ \ 3 / \ 3 /| \ 3 / \ 3 / / 1 \ | / 1 \ \ 3 / \ 3 /| \ 3 / \ 3 / 2022 \/ 2 2
- 2022*re|---------------------------------------------------| + I*|- 2022*im|---------------------------------------------------| + --------------------------------- + ----------------------------------| + --------------------------------- - ---------------------------------- + - 2022*re|---------------------------------------------------| + I*|- 2022*im|---------------------------------------------------| + --------------------------------- - ----------------------------------| + --------------------------------- + ---------------------------------- + - ----------------------------------- - -----------------------------------
| ______________________________| | | ______________________________| 2 2 | 2 2 | ______________________________| | | ______________________________| 2 2 | 2 2 ______________________________ 3
|/ ___\ / ______ ___ | | |/ ___\ / ______ ___ | | |/ ___\ / ______ ___ | | |/ ___\ / ______ ___ | | / ______ ___
|| 1 I*\/ 3 | / 27*\/ 2021 545535*I*\/ 3 | | || 1 I*\/ 3 | / 27*\/ 2021 545535*I*\/ 3 | | || 1 I*\/ 3 | / 27*\/ 2021 545535*I*\/ 3 | | || 1 I*\/ 3 | / 27*\/ 2021 545535*I*\/ 3 | | / 27*\/ 2021 545535*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
$$\left(\left(- \frac{\sqrt{2022} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} - 2022 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} + \frac{\sqrt{674} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + i \left(- 2022 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} + \frac{\sqrt{674} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + \frac{\sqrt{2022} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2}\right)\right) + \left(\frac{\sqrt{2022} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + \frac{\sqrt{674} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} - 2022 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} + i \left(- \frac{\sqrt{2022} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + \frac{\sqrt{674} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} - 2022 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)}\right)\right)\right) + \left(- \frac{\sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}{3} - \frac{2022}{\sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)$$
=
/ / / ______\\ / / ______\\\ / / / ______\\ / / ______\\\
______________________________ | | |20205*\/ 6063 || | |20205*\/ 6063 ||| | | |20205*\/ 6063 || | |20205*\/ 6063 |||
/ ______ ___ | |atan|--------------|| |atan|--------------||| | |atan|--------------|| |atan|--------------||| / / ______\\
/ 27*\/ 2021 545535*I*\/ 3 | _____ | \ 2021 /| ______ | \ 2021 /|| | _____ | \ 2021 /| ______ | \ 2021 /|| | |20205*\/ 6063 ||
3 / ----------- + -------------- | \/ 674 *sin|--------------------| \/ 2022 *cos|--------------------|| | \/ 674 *sin|--------------------| \/ 2022 *cos|--------------------|| |atan|--------------||
2022 / 1 \ / 1 \ \/ 2 2 | / 1 \ \ 3 / \ 3 /| | / 1 \ \ 3 / \ 3 /| _____ | \ 2021 /|
- ----------------------------------- - 2022*re|---------------------------------------------------| - 2022*re|---------------------------------------------------| - ----------------------------------- + I*|- 2022*im|---------------------------------------------------| + --------------------------------- - ----------------------------------| + I*|- 2022*im|---------------------------------------------------| + --------------------------------- + ----------------------------------| + \/ 674 *cos|--------------------|
______________________________ | ______________________________| | ______________________________| 3 | | ______________________________| 2 2 | | | ______________________________| 2 2 | \ 3 /
/ ______ ___ |/ ___\ / ______ ___ | |/ ___\ / ______ ___ | | |/ ___\ / ______ ___ | | | |/ ___\ / ______ ___ | |
/ 27*\/ 2021 545535*I*\/ 3 || 1 I*\/ 3 | / 27*\/ 2021 545535*I*\/ 3 | || 1 I*\/ 3 | / 27*\/ 2021 545535*I*\/ 3 | | || 1 I*\/ 3 | / 27*\/ 2021 545535*I*\/ 3 | | | || 1 I*\/ 3 | / 27*\/ 2021 545535*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 / /
$$- 2022 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} - 2022 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} + \sqrt{674} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)} - \frac{\sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}{3} + i \left(- 2022 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} + \frac{\sqrt{674} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + \frac{\sqrt{2022} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2}\right) + i \left(- \frac{\sqrt{2022} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + \frac{\sqrt{674} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} - 2022 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)}\right) - \frac{2022}{\sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}$$
произведение
/ / / / ______\\ / / ______\\\ / / ______\\ / / ______\\\ / / / / ______\\ / / ______\\\ / / ______\\ / / ______\\\ | | | |20205*\/ 6063 || | |20205*\/ 6063 ||| | |20205*\/ 6063 || | |20205*\/ 6063 ||| | | | |20205*\/ 6063 || | |20205*\/ 6063 ||| | |20205*\/ 6063 || | |20205*\/ 6063 ||| / ______________________________\ | | |atan|--------------|| |atan|--------------||| |atan|--------------|| |atan|--------------||| | | |atan|--------------|| |atan|--------------||| |atan|--------------|| |atan|--------------||| | / ______ ___ | | | _____ | \ 2021 /| ______ | \ 2021 /|| _____ | \ 2021 /| ______ | \ 2021 /|| | | _____ | \ 2021 /| ______ | \ 2021 /|| _____ | \ 2021 /| ______ | \ 2021 /|| | / 27*\/ 2021 545535*I*\/ 3 | | | \/ 674 *sin|--------------------| \/ 2022 *cos|--------------------|| \/ 674 *cos|--------------------| \/ 2022 *sin|--------------------|| | | \/ 674 *sin|--------------------| \/ 2022 *cos|--------------------|| \/ 674 *cos|--------------------| \/ 2022 *sin|--------------------|| | 3 / ----------- + -------------- | | / 1 \ | / 1 \ \ 3 / \ 3 /| \ 3 / \ 3 /| | / 1 \ | / 1 \ \ 3 / \ 3 /| \ 3 / \ 3 /| | 2022 \/ 2 2 | |- 2022*re|---------------------------------------------------| + I*|- 2022*im|---------------------------------------------------| + --------------------------------- + ----------------------------------| + --------------------------------- - ----------------------------------|*|- 2022*re|---------------------------------------------------| + I*|- 2022*im|---------------------------------------------------| + --------------------------------- - ----------------------------------| + --------------------------------- + ----------------------------------|*|- ----------------------------------- - -----------------------------------| | | ______________________________| | | ______________________________| 2 2 | 2 2 | | | ______________________________| | | ______________________________| 2 2 | 2 2 | | ______________________________ 3 | | |/ ___\ / ______ ___ | | |/ ___\ / ______ ___ | | | | |/ ___\ / ______ ___ | | |/ ___\ / ______ ___ | | | | / ______ ___ | | || 1 I*\/ 3 | / 27*\/ 2021 545535*I*\/ 3 | | || 1 I*\/ 3 | / 27*\/ 2021 545535*I*\/ 3 | | | | || 1 I*\/ 3 | / 27*\/ 2021 545535*I*\/ 3 | | || 1 I*\/ 3 | / 27*\/ 2021 545535*I*\/ 3 | | | | / 27*\/ 2021 545535*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 /
$$\left(- \frac{\sqrt{2022} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} - 2022 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} + \frac{\sqrt{674} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + i \left(- 2022 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} + \frac{\sqrt{674} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + \frac{\sqrt{2022} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2}\right)\right) \left(\frac{\sqrt{2022} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + \frac{\sqrt{674} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} - 2022 \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)} + i \left(- \frac{\sqrt{2022} \cos{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} + \frac{\sqrt{674} \sin{\left(\frac{\operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}}{2} - 2022 \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)}\right)\right) \left(- \frac{\sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}{3} - \frac{2022}{\sqrt[3]{\frac{27 \sqrt{2021}}{2} + \frac{545535 \sqrt{3} i}{2}}}\right)$$
=
/ / / ______\\\
| | |20205*\/ 6063 |||
| |2*atan|--------------||| / 2/3\
3 ___ | | \ 2021 /|| | 3 ___ / ______ ___\ |
337*\/ 2 *|1 - 2*cos|----------------------||*\1348 + \/ 2 *\\/ 2021 + 20205*I*\/ 3 / /
\ \ 3 //
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__________________________
3 / ______ ___
\/ \/ 2021 + 20205*I*\/ 3
$$\frac{337 \sqrt[3]{2} \left(1 - 2 \cos{\left(\frac{2 \operatorname{atan}{\left(\frac{20205 \sqrt{6063}}{2021} \right)}}{3} \right)}\right) \left(1348 + \sqrt[3]{2} \left(\sqrt{2021} + 20205 \sqrt{3} i\right)^{\frac{2}{3}}\right)}{\sqrt[3]{\sqrt{2021} + 20205 \sqrt{3} i}}$$
Теорема Виета
это приведённое кубическое уравнение
$$p x^{2} + q x + v + x^{3} = 0$$
где
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -2022$$
$$v = \frac{d}{a}$$
$$v = \sqrt{2021}$$
Формулы Виета
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -2022$$
$$x_{1} x_{2} x_{3} = \sqrt{2021}$$
$$p x^{2} + q x + v + x^{3} = 0$$
где
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -2022$$
$$v = \frac{d}{a}$$
$$v = \sqrt{2021}$$
Формулы Виета
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -2022$$
$$x_{1} x_{2} x_{3} = \sqrt{2021}$$