## Task B-2.1.

Determine for which value of the variable $x$ the expression $(3 a x+2015)^{2}+(2 a x+2015)^{2}, a \in \mathbb{R}$, $a \neq 0$ has the smallest value and what is the value of that minimum.
## Solution.

After squaring, we get

$$
\begin{aligned}
& 9 a^{2} x^{2}+6 a \cdot 2015 x+2015^{2}+4 a^{2} x^{2}+4 a \cdot 2015 x+2015^{2}= \\
& =13 a^{2} x^{2}+10 \cdot 2015 a x+2 \cdot 2015^{2}
\end{aligned}
$$

If the obtained expression is viewed as a quadratic function in the variable $x$, it has the smallest value for $x=-\frac{10 \cdot 2015 a}{2 \cdot 13 a^{2}}=-\frac{775}{a}$.

The smallest value of the given expression is

$$
\frac{4 \cdot 13 a^{2} \cdot 2 \cdot 2015^{2}-(10 \cdot 2015 a)^{2}}{4 \cdot 13 a^{2}}=\frac{4 a^{2} \cdot 2015^{2}}{4 \cdot 13 a^{2}}=\frac{2015^{2}}{13}=155 \cdot 2015=312325
$$

Note: If the student leaves the solution in the form $\frac{2015^{2}}{13}$, give full marks.