Example 28. Solve the equation

$$
\sqrt{2 x^{2}-1}+\sqrt{x^{2}-3 x-2}=\sqrt{2 x^{2}+2 x+3}+\sqrt{x^{2}-x+2}
$$
Solution. Rewrite equation (14) in such a way that after squaring both sides, we obtain an equivalent equation that is simpler than in other cases:

$$
\sqrt{2 x^{2}-1}-\sqrt{x^{2}-x+2}=\sqrt{2 x^{2}+2 x+3}-\sqrt{x^{2}-3 x-2}
$$

Considering that the left and right parts of the last equation must be simultaneously either non-negative or non-positive, and squaring both sides, we obtain an equivalent system consisting of an equation and a combination of two systems of inequalities:

$$
\left\{\begin{array} { l } 
{ 2 x ^ { 2 } - 1 - 2 \sqrt { 2 x ^ { 2 } - 1 } \sqrt { x ^ { 2 } - x + 2 } + x ^ { 2 } - x + 2 = } \\
{ = 2 x ^ { 2 } + 2 x + 3 - 2 \sqrt { 2 x ^ { 2 } + 2 x + 3 } \sqrt { x ^ { 2 } - 3 x - 2 } + x ^ { 2 } - 3 x - 2 }
\end{array} \left\{\begin{array}{l}
\left\{\begin{array}{l}
2 x^{2}-1 \geqslant x^{2}-x+2 \geqslant 0 \\
2 x^{2}+2 x+3 \geqslant x^{2}-3 x-2 \geqslant 0 \\
x^{2}-x+2 \geqslant 2 x^{2}-1 \geqslant 0 \\
x^{2}-3 x-2 \geqslant 2 x^{2}+2 x+3 \geqslant 0
\end{array}\right.
\end{array}\right.\right.
$$

which, after simplifications, takes the form

![](https://cdn.mathpix.com/cropped/2024_05_21_aca6fef227914c05e0acg-070.jpg?height=411&width=1142&top_left_y=1382&top_left_x=440)

This system is equivalent to a combination of two systems:

![](https://cdn.mathpix.com/cropped/2024_05_21_aca6fef227914c05e0acg-070.jpg?height=559&width=1114&top_left_y=1839&top_left_x=448)

We have:

$$
\begin{aligned}
&\left(2 x^{2}-1\right)\left(x^{2}-x+2\right)=\left(2 x^{2}+2 x+3\right)\left(x^{2}-3 x-2\right) \Leftrightarrow \\
& \Leftrightarrow x^{3}+5 x^{2}+7 x+2=0 \Leftrightarrow \\
& \Leftrightarrow(x+2)\left(x-\frac{-3+\sqrt{5}}{2}\right)\left(x-\frac{-3-\sqrt{5}}{2}\right)=0
\end{aligned}
$$

The roots \(x_{1}=-2\), \(x_{2}=\frac{-3+\sqrt{5}}{2}\), and \(x_{3}=\frac{-3-\sqrt{5}}{2}\) of the system (15) do not satisfy it, as \(x_{1}=-2\) does not satisfy the condition \(x^{2}+x-3 \geqslant 0\), \(x_{2}=\frac{\sqrt{5}-3}{2}\) does not satisfy the condition \(2 x^{2}-1 \geqslant 0\), and \(x_{3}=\frac{-\sqrt{5}-3}{2}\) does not satisfy the condition \(x^{2}+5 x+5 \geqslant 0\). Therefore, system (15) has no solutions.

The roots \(x_{2}=\frac{-3+\sqrt{5}}{2}\) and \(x_{3}=\frac{-3-\sqrt{\overline{5}}}{2}\) of the system (16) do not satisfy it, as \(x_{2}=\frac{\sqrt{\overline{5}}-3}{2}\) does not satisfy the condition \(2 x^{2}-1 \geqslant 0\), and \(x_{3}=\frac{-\sqrt{\overline{5}}-3}{2}\) does not satisfy the condition \(x^{2}+x-3 \leqslant 0\), while the root \(x_{1}=-2\) satisfies all conditions of system (16).

Thus, the combination of systems (15) and (16), and therefore equation (14), has a unique root \(x_{1}=-2\).