\[{\text{Решите уравнение }}x + \sqrt {{x^4} - 2x - 19} = 1\]
\[\sqrt {f\left( x \right)} = g\left( x \right) \Rightarrow {\text{ОДЗ: }}g\left( x \right) \ge 0\]
\[\begin{array}{l}
x + \sqrt {{x^4} - 2x - 19} = 1 \Leftrightarrow \sqrt {{x^4} - 2x - 19} = 1 - x\\
{\text{ОДЗ:}}\\
1 - x \ge 0 \Leftrightarrow x \le 1
\end{array}\]
\[\begin{array}{l}
{\text{Возводим уравнение }}\sqrt {{x^4} - 2x - 19} = 1 - x{\text{ в квадрат}}{\text{.}}\\
{x^4} - 2x - 19 = {\left( {1 - x} \right)^2} \Leftrightarrow \\
{x^4} - 2x - 19 = {x^2} - 2x + 1 \Leftrightarrow {x^4} - {x^2} - 20 = 0\\
{x^2} = t \Rightarrow {x^4} = {t^2}\\
{t^2} - t - 20 = 0 \Leftrightarrow \left\{ \begin{array}{l}
{t_1} + {t_2} = 1\\
{t_1} \cdot {t_2} = - 20
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
t = 5\\
t = - 4
\end{array} \right. \Rightarrow \\
\left[ \begin{array}{l}
{x^2} = 5\\
{x^2} = - 4
\end{array} \right. \Leftrightarrow x = \pm \sqrt 5
\end{array}\]
\[\begin{array}{l}
x = \sqrt 5 {\text{ не подходит по ОДЗ}} \Rightarrow \\
x = - \sqrt 5 {\text{ - единственное решение}}
\end{array}\]
\[x = - \sqrt 5 \]
