Solve the following equation:

$$
\sqrt{3 x+7}=3-\sqrt{5 x-26}
$$
I. Solution: Squaring both sides

$$
3 x+7=9-6 \sqrt{5 x-26}+5 x-26
$$

after rearranging and dividing by $(-2)$

$$
x-12=3 \sqrt{5 x-26}
$$

Squaring both sides again, and rearranging

$$
x^{2}-69 x+378=0
$$

If (1) has a solution, then it must also be a solution to (2), but the reverse is not true, because squaring is not an equivalent transformation and thus (2), which is only a consequence of (1), may contain extraneous roots.

Indeed, the roots of (2) are

$$
\left[x_{1}=63\right], \quad\left[x_{2}=6\right]
$$

neither of which satisfies (1), as we can verify by substitution $(14 \neq 3-17$ and $5 \neq 3-2)$. Therefore, our equation has no solution.

Note: We must not interpret our equation as

$$
\pm \sqrt{3 x+7}=3 \pm \sqrt{5 x-26}
$$

$$
\text { Ádám Antal (Bp. VIII., Széchenyi g. II. o. t.) }
$$

II. Solution: Our equation can also be written as:

$$
\sqrt{3 x+7}+\sqrt{5 x-26}=3
$$

The second term on the left side is only real if $x \geq \frac{26}{5}$, but then the first term, and thus the entire left side, is greater than 3. Therefore, our equation has no solution.