Please, I need a more detailed explanation of the particular solution of the problem with vectors Here is the problem and its solution (link to the source if you are interested):
Two different points $A$ and $B$ are given. Find a set of such points $M$, that $\overrightarrow{MA}\cdot\overrightarrow{MB}=k^2$, where $k$ is a given non-zero number.
The solution:
Let coordinates of points be: $A(x_a, y_a), B(x_b, y_b), M(x, y)$.
Then: 
$$
\begin{align}
& \overrightarrow{MA}=(x_a - x, y_a - y), \overrightarrow{MB}=(x_b - x, y_b - y), \\ 
& \overrightarrow{MA}\cdot\overrightarrow{MB}= (x_a - x)(x_b - x)+(y_a - y)(y_b - y)=k^2
\end{align}
$$
Transform the last expression, opening brackets:
$$
x_a x_b - (x_a+x_b)x + x^2 + y_a y_b - (y_a+y_b)y + y^2 = k^2
$$
At this point I ask someone to provide a more detailed explanation, please, cause I failed to understand the next steps. What a transformation has been made here?
$$
\left( x - \frac{x_a + x_b}{2} \right)^2 - \frac{(x_a - x_b)^2}{4} + 
\left( y - \frac{y_a + y_b}{2} \right)^2 - \frac{(y_a - y_b)^2}{4} = k^2 \\
\left( x - \frac{x_a + x_b}{2} \right)^2 + \left( y - \frac{y_a + y_b}{2} \right)^2 
 = k^2 + \frac{(x_a - x_b)^2}{4} + \frac{(y_a - y_b)^2}{4}
$$
Thus, this is a circumference with a center at the middle of segment $AB$ and its radius. (This I also failed to understand)
$$
r = \sqrt{k^2 + \frac{|AB|^2}{4}}
$$

Hints:
$$x^2\pm a=\left(x\pm\frac{a}{2}\right)^2-\frac{a^2}{4}$$
and this is what they do in the first step, with both $\,x\,,\,y\,$ . For example,
$$y^2-(y_a+y_b)y=\left(y-\frac{y_a+y_b}{2}\right)^2-\frac{(y_a+y_b)^2}{4}\;,\;\;etc.$$
$$|AB|^2=(x_a-x_b)^2+(y_a-y_b)^2\;\ldots$$
