Solve the following equation:

$$
\sqrt{2 x+8}-\sqrt{x+5}=7 .
$$
$$
\sqrt{2 x+8}=7+\sqrt{x+5}:
$$

Squaring and simplifying

$$
x-46=14 \sqrt{x+5}
$$

Squaring again

$$
x^{2}-92 x+2116=196 x+980
$$

which is

$$
x^{2}-288 x+1136=0
$$

from which

$$
x_{1}=284\left[x_{2}=4\right]
$$

Since the roots of the equation obtained by squaring may include those that do not satisfy the original equation (it is only true that every root of the original equation satisfies the equation obtained by squaring, but not vice versa), it must be checked by substitution which roots satisfy the original equation as well. In this case, we find that 284 satisfies $(\sqrt{576}-\sqrt{289}=24-17=7)$, while 4 does not satisfy our equation.

$$
(\sqrt{16}-\sqrt{9}=4-3 \neq 7 . \quad \text { However, } 4 \text { would satisfy the } \sqrt{2 x+8}+\sqrt{x+5}=7
$$

equation, which 284 does not satisfy.)

Magyar Károly (Debrecen, 3rd grade chemical engineering technology II. class)