2^x-x-1=8 (уравнение)

Найду корень уравнения: 2^x-x-1=8

Вы ввели

 x            
2  - x - 1 = 8

$$\left(2^{x} - x\right) - 1 = 8$$

Быстрый ответ

           /-log(2) \
          W|--------|
           \  512   /
x1 = -9 - -----------
             log(2)  

$$x_{1} = -9 - \frac{W\left(- \frac{\log{\left(2 \right)}}{512}\right)}{\log{\left(2 \right)}}$$

           /-log(2)     \
          W|--------, -1|
           \  512       /
x2 = -9 - ---------------
               log(2)    

$$x_{2} = -9 - \frac{W_{-1}\left(- \frac{\log{\left(2 \right)}}{512}\right)}{\log{\left(2 \right)}}$$

Сумма и произведение корней

сумма

      /-log(2) \         /-log(2)     \
     W|--------|        W|--------, -1|
      \  512   /         \  512       /
-9 - ----------- + -9 - ---------------
        log(2)               log(2)    

$$\left(-9 - \frac{W\left(- \frac{\log{\left(2 \right)}}{512}\right)}{\log{\left(2 \right)}}\right) + \left(-9 - \frac{W_{-1}\left(- \frac{\log{\left(2 \right)}}{512}\right)}{\log{\left(2 \right)}}\right)$$

=

       /-log(2) \    /-log(2)     \
      W|--------|   W|--------, -1|
       \  512   /    \  512       /
-18 - ----------- - ---------------
         log(2)          log(2)    

$$-18 - \frac{W\left(- \frac{\log{\left(2 \right)}}{512}\right)}{\log{\left(2 \right)}} - \frac{W_{-1}\left(- \frac{\log{\left(2 \right)}}{512}\right)}{\log{\left(2 \right)}}$$

произведение

/      /-log(2) \\ /      /-log(2)     \\
|     W|--------|| |     W|--------, -1||
|      \  512   /| |      \  512       /|
|-9 - -----------|*|-9 - ---------------|
\        log(2)  / \          log(2)    /

$$\left(-9 - \frac{W\left(- \frac{\log{\left(2 \right)}}{512}\right)}{\log{\left(2 \right)}}\right) \left(-9 - \frac{W_{-1}\left(- \frac{\log{\left(2 \right)}}{512}\right)}{\log{\left(2 \right)}}\right)$$

=

/            /-log(2) \\ /            /-log(2)     \\
|9*log(2) + W|--------||*|9*log(2) + W|--------, -1||
\            \  512   // \            \  512       //
-----------------------------------------------------
                          2                          
                       log (2)                       

$$\frac{\left(W\left(- \frac{\log{\left(2 \right)}}{512}\right) + 9 \log{\left(2 \right)}\right) \left(W_{-1}\left(- \frac{\log{\left(2 \right)}}{512}\right) + 9 \log{\left(2 \right)}\right)}{\log{\left(2 \right)}^{2}}$$

Численный ответ

x1 = -8.99804422547091

x2 = 3.6624891248601

График