Example 7. Solve the equation

$$
\sqrt{2 x^{2}+5 x-2}-\sqrt{2 x^{2}+5 x-9}=1
$$
Solution. First method. By direct verification, we find that the following relation holds:

$$
\left(\sqrt{2 x^{2}+5 x-2}\right)^{2}-\left(\sqrt{2 x^{2}+5 x-9}\right)^{2}=7
$$

We factorize the left-hand side of this equation:

$$
\begin{aligned}
&\left(\sqrt{2 x^{2}+5 x-2}-\sqrt{2 x^{2}+5 x-9}\right) \times \\
& \times\left(\sqrt{2 x^{2}+5 x-2}+\sqrt{2 x^{2}+5 x-9}\right)=7
\end{aligned}
$$

By replacing the difference with one, we obtain the equation

$$
\sqrt{2 x^{2}+5 x-2}+\sqrt{2 x^{2}+5 x-9}=7
$$

Adding the left and right parts of the obtained and original equations, we get the equation

$$
\sqrt{2 x^{2}+5 x-2}=4
$$

which is a consequence of the original equation and has roots \(x_{1}=2, x_{2}=-9 / 2\).

Direct substitution of these numbers into the original equation shows that both are solutions of the original equation.

In the second method. Let \(2 x^{2}+5 x-2=t\). Then the original equation becomes \(\sqrt{t}=\sqrt{t-7}+1\). Squaring both sides of the obtained equation and combining like terms, we get the equation \(\sqrt{t-7}=3\), which is a consequence of the previous one. From this, we find \(t=16\).

Returning to the unknown \(x\), we get the equation

$$
2 x^{2}+5 x-2=16
$$

which is a consequence of the original equation. By verification, we find that its roots \(x_{1}=2\) and \(x_{2}=-9 / 2\) are roots of the original equation.