$3+$ [Sums of numerical sequences and series of differences]

Solve the equation $(x+1)^{2}+(x+2)^{2}+\ldots+(x+10)^{2}=(x+1+2+\ldots+10)^{2}$.
Let's square and group some terms:

$10 x^{2}+2(1+2+\ldots+10) x+1^{2}+2^{2}+\ldots+10^{2}=x^{2}+2(1+2+\ldots+10) x+(1+2+\ldots+10)^{2}$.

Considering that $1^{2}+2^{2}+\ldots+10^{2}=1 / 6 \cdot 10 \cdot 11 \cdot 21=385$, and $(1+2+\ldots+10)^{2}=55^{2}=3025$, after combining like terms, we get

$9 x^{2}=2640$, that is, $x= \pm \frac{4 \sqrt{165}}{3}$.

## Answer

$$
\pm \frac{4 \sqrt{165}}{3}
$$