Đáp án:
\({\left[ {\begin{array}{*{20}{l}}
{x = 1}\\
{x = 1,906795303}
\end{array}} \right.}\)
Giải thích các bước giải:
\(\begin{array}{*{20}{l}}
{DK:\left\{ {\begin{array}{*{20}{l}}
{ - {x^2} + 3x - 2 \ge 0}\\
{x + 1 \ge 0}
\end{array}} \right.}\\
{ \to \left\{ {\begin{array}{*{20}{l}}
{1 \le x \le 2}\\
{x \ge {\rm{ \;}} - 1}
\end{array}} \right.}\\
{ \to 1 \le x \le 2}\\
{\sqrt { - {x^2} + 3x - 2} {\rm{ \;}} + \sqrt {x + 1} {\rm{ \;}} = \sqrt 2 }\\
{{\rm{\;}} \to \sqrt { - {x^2} + 3x - 2} {\rm{ \;}} = \sqrt 2 {\rm{ \;}} - \sqrt {x + 1} }\\
{ \to {\rm{ \;}} - {x^2} + 3x - 2 = 2 - 2\sqrt {2x + 2} {\rm{ \;}} + x + 1}\\
{ \to {\rm{ \;}} - {x^2} + 2x - 5 = {\rm{ \;}} - 2\sqrt {2x + 2} }\\
{ \to {x^2} - 2x + 5 = 2\sqrt {2x + 2} }\\
{ \to {x^4} + 4{x^2} + 25 - 4{x^3} + 10{x^2} - 20x = 4\left( {2x + 2} \right)}\\
{ \to {x^4} - 4{x^3} + 14{x^2} - 28x + 17 = 0}\\
{ \to {x^4} - {x^3} - 3{x^3} + 3{x^2} + 11{x^2} - 11x - 17x + 17 = 0}\\
{ \to {x^3}\left( {x - 1} \right) - 3{x^2}\left( {x - 1} \right) + 11x\left( {x - 1} \right) - 17\left( {x - 1} \right) = 0}\\
{ \to \left( {x - 1} \right)\left( {{x^3} - 3{x^2} + 11x - 17} \right) = 0}\\
{ \to \left[ {\begin{array}{*{20}{l}}
{x = 1\left( {TM} \right)}\\
{{x^3} - 3{x^2} + 11x - 17 = 0}
\end{array}} \right.}\\
{ \to \left[ {\begin{array}{*{20}{l}}
{x = 1}\\
{x = 1,906795303\left( {TM} \right)}
\end{array}} \right.}\\
{KL:\left[ {\begin{array}{*{20}{l}}
{x = 1}\\
{x = 1,906795303}
\end{array}} \right.}
\end{array}\)
