2*x^3-6*x^2+1=0 (уравнение)

Найду корень уравнения: 2*x^3-6*x^2+1=0

Вы ввели

   3      2        
2*x  - 6*x  + 1 = 0

$$\left(2 x^{3} - 6 x^{2}\right) + 1 = 0$$

Быстрый ответ

            /    /  ___\\     /     /    /  ___\\            /    /  ___\\                                         \            /    /  ___\\                                         
            |    |\/ 7 ||     |     |    |\/ 7 ||            |    |\/ 7 ||                                         |            |    |\/ 7 ||                                         
            |atan|-----||     |     |atan|-----||            |atan|-----||                                         |            |atan|-----||                                         
            |    \  3  /|     |     |    \  3  /|     ___    |    \  3  /|                                         |     ___    |    \  3  /|                                         
         cos|-----------|     |  sin|-----------|   \/ 3 *cos|-----------|                                         |   \/ 3 *sin|-----------|                                         
            \     3     /     |     \     3     /            \     3     /     /                1                 \|            \     3     /     /                1                 \
x1 = 1 - ---------------- + I*|- ---------------- - ---------------------- + im|----------------------------------|| + ---------------------- + re|----------------------------------|
                2             |         2                     2                |                     _____________||             2                |                     _____________|
                              |                                                |/          ___\     /         ___ ||                              |/          ___\     /         ___ |
                              |                                                ||  1   I*\/ 3 |    /  3   I*\/ 7  ||                              ||  1   I*\/ 3 |    /  3   I*\/ 7  |
                              |                                                ||- - - -------|*3 /   - + ------- ||                              ||- - - -------|*3 /   - + ------- |
                              \                                                \\  2      2   / \/    4      4    //                              \\  2      2   / \/    4      4    /

$$x_{1} = - \frac{\cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)} + \frac{\sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + 1 + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} - \frac{\sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)}\right)$$

            /    /  ___\\     /     /    /  ___\\            /    /  ___\\                                         \            /    /  ___\\                                         
            |    |\/ 7 ||     |     |    |\/ 7 ||            |    |\/ 7 ||                                         |            |    |\/ 7 ||                                         
            |atan|-----||     |     |atan|-----||            |atan|-----||                                         |            |atan|-----||                                         
            |    \  3  /|     |     |    \  3  /|     ___    |    \  3  /|                                         |     ___    |    \  3  /|                                         
         cos|-----------|     |  sin|-----------|   \/ 3 *cos|-----------|                                         |   \/ 3 *sin|-----------|                                         
            \     3     /     |     \     3     /            \     3     /     /                1                 \|            \     3     /     /                1                 \
x2 = 1 - ---------------- + I*|- ---------------- + ---------------------- + im|----------------------------------|| - ---------------------- + re|----------------------------------|
                2             |         2                     2                |                     _____________||             2                |                     _____________|
                              |                                                |/          ___\     /         ___ ||                              |/          ___\     /         ___ |
                              |                                                ||  1   I*\/ 3 |    /  3   I*\/ 7  ||                              ||  1   I*\/ 3 |    /  3   I*\/ 7  |
                              |                                                ||- - + -------|*3 /   - + ------- ||                              ||- - + -------|*3 /   - + ------- |
                              \                                                \\  2      2   / \/    4      4    //                              \\  2      2   / \/    4      4    /

$$x_{2} = \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)} - \frac{\cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} - \frac{\sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + 1 + i \left(\operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)} - \frac{\sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2}\right)$$

                                   _____________
                                  /         ___ 
                 1               /  3   I*\/ 7  
x3 = 1 + ------------------ + 3 /   - + ------- 
              _____________   \/    4      4    
             /         ___                      
            /  3   I*\/ 7                       
         3 /   - + -------                      
         \/    4      4                         

$$x_{3} = 1 + \frac{1}{\sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}} + \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}$$

Сумма и произведение корней

сумма

       /    /  ___\\     /     /    /  ___\\            /    /  ___\\                                         \            /    /  ___\\                                                   /    /  ___\\     /     /    /  ___\\            /    /  ___\\                                         \            /    /  ___\\                                                                                       
       |    |\/ 7 ||     |     |    |\/ 7 ||            |    |\/ 7 ||                                         |            |    |\/ 7 ||                                                   |    |\/ 7 ||     |     |    |\/ 7 ||            |    |\/ 7 ||                                         |            |    |\/ 7 ||                                                                                       
       |atan|-----||     |     |atan|-----||            |atan|-----||                                         |            |atan|-----||                                                   |atan|-----||     |     |atan|-----||            |atan|-----||                                         |            |atan|-----||                                                                                       
       |    \  3  /|     |     |    \  3  /|     ___    |    \  3  /|                                         |     ___    |    \  3  /|                                                   |    \  3  /|     |     |    \  3  /|     ___    |    \  3  /|                                         |     ___    |    \  3  /|                                                                          _____________
    cos|-----------|     |  sin|-----------|   \/ 3 *cos|-----------|                                         |   \/ 3 *sin|-----------|                                                cos|-----------|     |  sin|-----------|   \/ 3 *cos|-----------|                                         |   \/ 3 *sin|-----------|                                                                         /         ___ 
       \     3     /     |     \     3     /            \     3     /     /                1                 \|            \     3     /     /                1                 \          \     3     /     |     \     3     /            \     3     /     /                1                 \|            \     3     /     /                1                 \               1               /  3   I*\/ 7  
1 - ---------------- + I*|- ---------------- - ---------------------- + im|----------------------------------|| + ---------------------- + re|----------------------------------| + 1 - ---------------- + I*|- ---------------- + ---------------------- + im|----------------------------------|| - ---------------------- + re|----------------------------------| + 1 + ------------------ + 3 /   - + ------- 
           2             |         2                     2                |                     _____________||             2                |                     _____________|              2             |         2                     2                |                     _____________||             2                |                     _____________|            _____________   \/    4      4    
                         |                                                |/          ___\     /         ___ ||                              |/          ___\     /         ___ |                            |                                                |/          ___\     /         ___ ||                              |/          ___\     /         ___ |           /         ___                      
                         |                                                ||  1   I*\/ 3 |    /  3   I*\/ 7  ||                              ||  1   I*\/ 3 |    /  3   I*\/ 7  |                            |                                                ||  1   I*\/ 3 |    /  3   I*\/ 7  ||                              ||  1   I*\/ 3 |    /  3   I*\/ 7  |          /  3   I*\/ 7                       
                         |                                                ||- - - -------|*3 /   - + ------- ||                              ||- - - -------|*3 /   - + ------- |                            |                                                ||- - + -------|*3 /   - + ------- ||                              ||- - + -------|*3 /   - + ------- |       3 /   - + -------                      
                         \                                                \\  2      2   / \/    4      4    //                              \\  2      2   / \/    4      4    /                            \                                                \\  2      2   / \/    4      4    //                              \\  2      2   / \/    4      4    /       \/    4      4                         

$$\left(\left(\operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)} - \frac{\cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} - \frac{\sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + 1 + i \left(\operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)} - \frac{\sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2}\right)\right) + \left(- \frac{\cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)} + \frac{\sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + 1 + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} - \frac{\sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)}\right)\right)\right) + \left(1 + \frac{1}{\sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}} + \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}\right)$$

=

                                                                   /     /    /  ___\\            /    /  ___\\                                         \     /     /    /  ___\\            /    /  ___\\                                         \                                                                                  
                                                                   |     |    |\/ 7 ||            |    |\/ 7 ||                                         |     |     |    |\/ 7 ||            |    |\/ 7 ||                                         |                                                                                  
                                                 /    /  ___\\     |     |atan|-----||            |atan|-----||                                         |     |     |atan|-----||            |atan|-----||                                         |                                                                                  
                              _____________      |    |\/ 7 ||     |     |    \  3  /|     ___    |    \  3  /|                                         |     |     |    \  3  /|     ___    |    \  3  /|                                         |                                                                                  
                             /         ___       |atan|-----||     |  sin|-----------|   \/ 3 *cos|-----------|                                         |     |  sin|-----------|   \/ 3 *cos|-----------|                                         |                                                                                  
            1               /  3   I*\/ 7        |    \  3  /|     |     \     3     /            \     3     /     /                1                 \|     |     \     3     /            \     3     /     /                1                 \|     /                1                 \     /                1                 \
3 + ------------------ + 3 /   - + -------  - cos|-----------| + I*|- ---------------- + ---------------------- + im|----------------------------------|| + I*|- ---------------- - ---------------------- + im|----------------------------------|| + re|----------------------------------| + re|----------------------------------|
         _____________   \/    4      4          \     3     /     |         2                     2                |                     _____________||     |         2                     2                |                     _____________||     |                     _____________|     |                     _____________|
        /         ___                                              |                                                |/          ___\     /         ___ ||     |                                                |/          ___\     /         ___ ||     |/          ___\     /         ___ |     |/          ___\     /         ___ |
       /  3   I*\/ 7                                               |                                                ||  1   I*\/ 3 |    /  3   I*\/ 7  ||     |                                                ||  1   I*\/ 3 |    /  3   I*\/ 7  ||     ||  1   I*\/ 3 |    /  3   I*\/ 7  |     ||  1   I*\/ 3 |    /  3   I*\/ 7  |
    3 /   - + -------                                              |                                                ||- - + -------|*3 /   - + ------- ||     |                                                ||- - - -------|*3 /   - + ------- ||     ||- - + -------|*3 /   - + ------- |     ||- - - -------|*3 /   - + ------- |
    \/    4      4                                                 \                                                \\  2      2   / \/    4      4    //     \                                                \\  2      2   / \/    4      4    //     \\  2      2   / \/    4      4    /     \\  2      2   / \/    4      4    /

$$- \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)} + 3 + \frac{1}{\sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}} + i \left(\operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)} - \frac{\sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2}\right) + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} - \frac{\sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)}\right) + \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}$$

произведение

/       /    /  ___\\     /     /    /  ___\\            /    /  ___\\                                         \            /    /  ___\\                                         \ /       /    /  ___\\     /     /    /  ___\\            /    /  ___\\                                         \            /    /  ___\\                                         \                                              
|       |    |\/ 7 ||     |     |    |\/ 7 ||            |    |\/ 7 ||                                         |            |    |\/ 7 ||                                         | |       |    |\/ 7 ||     |     |    |\/ 7 ||            |    |\/ 7 ||                                         |            |    |\/ 7 ||                                         |                                              
|       |atan|-----||     |     |atan|-----||            |atan|-----||                                         |            |atan|-----||                                         | |       |atan|-----||     |     |atan|-----||            |atan|-----||                                         |            |atan|-----||                                         |                                              
|       |    \  3  /|     |     |    \  3  /|     ___    |    \  3  /|                                         |     ___    |    \  3  /|                                         | |       |    \  3  /|     |     |    \  3  /|     ___    |    \  3  /|                                         |     ___    |    \  3  /|                                         | /                              _____________\
|    cos|-----------|     |  sin|-----------|   \/ 3 *cos|-----------|                                         |   \/ 3 *sin|-----------|                                         | |    cos|-----------|     |  sin|-----------|   \/ 3 *cos|-----------|                                         |   \/ 3 *sin|-----------|                                         | |                             /         ___ |
|       \     3     /     |     \     3     /            \     3     /     /                1                 \|            \     3     /     /                1                 \| |       \     3     /     |     \     3     /            \     3     /     /                1                 \|            \     3     /     /                1                 \| |            1               /  3   I*\/ 7  |
|1 - ---------------- + I*|- ---------------- - ---------------------- + im|----------------------------------|| + ---------------------- + re|----------------------------------||*|1 - ---------------- + I*|- ---------------- + ---------------------- + im|----------------------------------|| - ---------------------- + re|----------------------------------||*|1 + ------------------ + 3 /   - + ------- |
|           2             |         2                     2                |                     _____________||             2                |                     _____________|| |           2             |         2                     2                |                     _____________||             2                |                     _____________|| |         _____________   \/    4      4    |
|                         |                                                |/          ___\     /         ___ ||                              |/          ___\     /         ___ || |                         |                                                |/          ___\     /         ___ ||                              |/          ___\     /         ___ || |        /         ___                      |
|                         |                                                ||  1   I*\/ 3 |    /  3   I*\/ 7  ||                              ||  1   I*\/ 3 |    /  3   I*\/ 7  || |                         |                                                ||  1   I*\/ 3 |    /  3   I*\/ 7  ||                              ||  1   I*\/ 3 |    /  3   I*\/ 7  || |       /  3   I*\/ 7                       |
|                         |                                                ||- - - -------|*3 /   - + ------- ||                              ||- - - -------|*3 /   - + ------- || |                         |                                                ||- - + -------|*3 /   - + ------- ||                              ||- - + -------|*3 /   - + ------- || |    3 /   - + -------                      |
\                         \                                                \\  2      2   / \/    4      4    //                              \\  2      2   / \/    4      4    // \                         \                                                \\  2      2   / \/    4      4    //                              \\  2      2   / \/    4      4    // \    \/    4      4                         /

$$\left(- \frac{\cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + \operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)} + \frac{\sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + 1 + i \left(- \frac{\sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} - \frac{\sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + \operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} - \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)}\right)\right) \left(\operatorname{re}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)} - \frac{\cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} - \frac{\sqrt{3} \sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + 1 + i \left(\operatorname{im}{\left(\frac{1}{\left(- \frac{1}{2} + \frac{\sqrt{3} i}{2}\right) \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}}\right)} - \frac{\sin{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2} + \frac{\sqrt{3} \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}}{2}\right)\right) \left(1 + \frac{1}{\sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}} + \sqrt[3]{\frac{3}{4} + \frac{\sqrt{7} i}{4}}\right)$$

=

/     /    /  ___\\      /      /  ___\\\                                                     
|     |    |\/ 7 ||      |      |\/ 7 |||                                                     
|     |atan|-----||      |2*atan|-----||| /            _____________ /       _______________\\
|     |    \  3  /|      |      \  3  /|| |   2/3   3 /         ___  |    3 /           ___ ||
|- cos|-----------| + cos|-------------||*\2*2    + \/  3 + I*\/ 7  *\2 + \/  6 + 2*I*\/ 7  //
\     \     3     /      \      3      //                                                     
----------------------------------------------------------------------------------------------
                                          _____________                                       
                                       3 /         ___                                        
                                       \/  3 + I*\/ 7                                         

$$\frac{\left(2 \cdot 2^{\frac{2}{3}} + \left(2 + \sqrt[3]{6 + 2 \sqrt{7} i}\right) \sqrt[3]{3 + \sqrt{7} i}\right) \left(- \cos{\left(\frac{\operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)} + \cos{\left(\frac{2 \operatorname{atan}{\left(\frac{\sqrt{7}}{3} \right)}}{3} \right)}\right)}{\sqrt[3]{3 + \sqrt{7} i}}$$

Теорема Виета

перепишем уравнение
$$\left(2 x^{3} - 6 x^{2}\right) + 1 = 0$$
из
$$a x^{3} + b x^{2} + c x + d = 0$$
как приведённое кубическое уравнение
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - 3 x^{2} + \frac{1}{2} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
где
$$p = \frac{b}{a}$$
$$p = -3$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = \frac{1}{2}$$
Формулы Виета
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 3$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = \frac{1}{2}$$

Численный ответ

x1 = 0.442125301668475

x2 = -0.384367152638142

x3 = 2.94224185096967

График