sqrt(x+1)=x+a (уравнение)

Найду корень уравнения: sqrt(x+1)=x+a

Вы ввели

  _______        
\/ x + 1  = x + a

$$\sqrt{x + 1} = a + x$$

Упростить
Выразить через переменную a
График неявной функции

Подробное решение

Дано уравнение
$$\sqrt{x + 1} = a + x$$
$$\sqrt{x + 1} = a + x$$
Возведём обе части ур-ния в(о) 2-ую степень
$$x + 1 = \left(a + x\right)^{2}$$
$$x + 1 = a^{2} + 2 a x + x^{2}$$
Перенесём правую часть уравнения левую часть уравнения со знаком минус
$$- a^{2} - 2 a x - x^{2} + x + 1 = 0$$
Это уравнение вида

a*x^2 + b*x + c = 0

Квадратное уравнение можно решить
с помощью дискриминанта.
Корни квадратного уравнения:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
где D = b^2 - 4*a*c - это дискриминант.
Т.к.
$$a = -1$$
$$b = 1 - 2 a$$
$$c = 1 - a^{2}$$
, то

D = b^2 - 4 * a * c = 

(1 - 2*a)^2 - 4 * (-1) * (1 - a^2) = 4 + (1 - 2*a)^2 - 4*a^2

Уравнение имеет два корня.

x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

или
$$x_{1} = - a - \frac{\sqrt{- 4 a^{2} + \left(1 - 2 a\right)^{2} + 4}}{2} + \frac{1}{2}$$
Упростить
$$x_{2} = - a + \frac{\sqrt{- 4 a^{2} + \left(1 - 2 a\right)^{2} + 4}}{2} + \frac{1}{2}$$
Упростить

Быстрый ответ

                   /            ____________________________                                  \      ____________________________                                  
                   |         4 /              2        2        /atan2(-4*im(a), 5 - 4*re(a))\|   4 /              2        2        /atan2(-4*im(a), 5 - 4*re(a))\
                   |         \/  (5 - 4*re(a))  + 16*im (a) *sin|----------------------------||   \/  (5 - 4*re(a))  + 16*im (a) *cos|----------------------------|
     1             |                                            \             2              /|                                      \             2              /
x1 = - - re(a) + I*|-im(a) - -----------------------------------------------------------------| - -----------------------------------------------------------------
     2             \                                         2                                /                                   2                                

$$x_{1} = i \left(- \frac{\sqrt[4]{\left(5 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},5 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \operatorname{im}{\left(a\right)}\right) - \frac{\sqrt[4]{\left(5 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},5 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \operatorname{re}{\left(a\right)} + \frac{1}{2}$$

                   /            ____________________________                                  \      ____________________________                                  
                   |         4 /              2        2        /atan2(-4*im(a), 5 - 4*re(a))\|   4 /              2        2        /atan2(-4*im(a), 5 - 4*re(a))\
                   |         \/  (5 - 4*re(a))  + 16*im (a) *sin|----------------------------||   \/  (5 - 4*re(a))  + 16*im (a) *cos|----------------------------|
     1             |                                            \             2              /|                                      \             2              /
x2 = - - re(a) + I*|-im(a) + -----------------------------------------------------------------| + -----------------------------------------------------------------
     2             \                                         2                                /                                   2                                

$$x_{2} = i \left(\frac{\sqrt[4]{\left(5 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},5 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \operatorname{im}{\left(a\right)}\right) + \frac{\sqrt[4]{\left(5 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},5 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \operatorname{re}{\left(a\right)} + \frac{1}{2}$$

Сумма и произведение корней

сумма

              /            ____________________________                                  \      ____________________________                                                   /            ____________________________                                  \      ____________________________                                  
              |         4 /              2        2        /atan2(-4*im(a), 5 - 4*re(a))\|   4 /              2        2        /atan2(-4*im(a), 5 - 4*re(a))\                 |         4 /              2        2        /atan2(-4*im(a), 5 - 4*re(a))\|   4 /              2        2        /atan2(-4*im(a), 5 - 4*re(a))\
              |         \/  (5 - 4*re(a))  + 16*im (a) *sin|----------------------------||   \/  (5 - 4*re(a))  + 16*im (a) *cos|----------------------------|                 |         \/  (5 - 4*re(a))  + 16*im (a) *sin|----------------------------||   \/  (5 - 4*re(a))  + 16*im (a) *cos|----------------------------|
1             |                                            \             2              /|                                      \             2              /   1             |                                            \             2              /|                                      \             2              /
- - re(a) + I*|-im(a) - -----------------------------------------------------------------| - ----------------------------------------------------------------- + - - re(a) + I*|-im(a) + -----------------------------------------------------------------| + -----------------------------------------------------------------
2             \                                         2                                /                                   2                                   2             \                                         2                                /                                   2                                

$$\left(i \left(- \frac{\sqrt[4]{\left(5 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},5 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \operatorname{im}{\left(a\right)}\right) - \frac{\sqrt[4]{\left(5 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},5 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \operatorname{re}{\left(a\right)} + \frac{1}{2}\right) + \left(i \left(\frac{\sqrt[4]{\left(5 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},5 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \operatorname{im}{\left(a\right)}\right) + \frac{\sqrt[4]{\left(5 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},5 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \operatorname{re}{\left(a\right)} + \frac{1}{2}\right)$$

=

                /            ____________________________                                  \     /            ____________________________                                  \
                |         4 /              2        2        /atan2(-4*im(a), 5 - 4*re(a))\|     |         4 /              2        2        /atan2(-4*im(a), 5 - 4*re(a))\|
                |         \/  (5 - 4*re(a))  + 16*im (a) *sin|----------------------------||     |         \/  (5 - 4*re(a))  + 16*im (a) *sin|----------------------------||
                |                                            \             2              /|     |                                            \             2              /|
1 - 2*re(a) + I*|-im(a) + -----------------------------------------------------------------| + I*|-im(a) - -----------------------------------------------------------------|
                \                                         2                                /     \                                         2                                /

$$i \left(- \frac{\sqrt[4]{\left(5 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},5 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \operatorname{im}{\left(a\right)}\right) + i \left(\frac{\sqrt[4]{\left(5 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},5 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \operatorname{im}{\left(a\right)}\right) - 2 \operatorname{re}{\left(a\right)} + 1$$

произведение

/              /            ____________________________                                  \      ____________________________                                  \ /              /            ____________________________                                  \      ____________________________                                  \
|              |         4 /              2        2        /atan2(-4*im(a), 5 - 4*re(a))\|   4 /              2        2        /atan2(-4*im(a), 5 - 4*re(a))\| |              |         4 /              2        2        /atan2(-4*im(a), 5 - 4*re(a))\|   4 /              2        2        /atan2(-4*im(a), 5 - 4*re(a))\|
|              |         \/  (5 - 4*re(a))  + 16*im (a) *sin|----------------------------||   \/  (5 - 4*re(a))  + 16*im (a) *cos|----------------------------|| |              |         \/  (5 - 4*re(a))  + 16*im (a) *sin|----------------------------||   \/  (5 - 4*re(a))  + 16*im (a) *cos|----------------------------||
|1             |                                            \             2              /|                                      \             2              /| |1             |                                            \             2              /|                                      \             2              /|
|- - re(a) + I*|-im(a) - -----------------------------------------------------------------| - -----------------------------------------------------------------|*|- - re(a) + I*|-im(a) + -----------------------------------------------------------------| + -----------------------------------------------------------------|
\2             \                                         2                                /                                   2                                / \2             \                                         2                                /                                   2                                /

$$\left(i \left(- \frac{\sqrt[4]{\left(5 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},5 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \operatorname{im}{\left(a\right)}\right) - \frac{\sqrt[4]{\left(5 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},5 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \operatorname{re}{\left(a\right)} + \frac{1}{2}\right) \left(i \left(\frac{\sqrt[4]{\left(5 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},5 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \operatorname{im}{\left(a\right)}\right) + \frac{\sqrt[4]{\left(5 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},5 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \operatorname{re}{\left(a\right)} + \frac{1}{2}\right)$$

=

       2        2                     
-1 + re (a) - im (a) + 2*I*im(a)*re(a)

$$\left(\operatorname{re}{\left(a\right)}\right)^{2} + 2 i \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2} - 1$$