<s>[INST] Why is $x^2+x$ the same as writing $(x+0.5)^2-0.25$? I find it extremely weird that a translation vertically by a variable should cause the graph to move both vertically and horizontally. Also, why 0.5 and 0.25?
 [/INST] Consider a more general case: ($a > 0$)
\begin{align}
a \, x^2 + b \, x &= (\sqrt{a})^2 x^2 + 2 \cdot \frac{b}{2 \, \sqrt{a}} \cdot \sqrt{a} x + \frac{b^2}{4 \, a} - \frac{b^2}{4 \, a} \\
&= \left(\sqrt{a} x + \frac{b}{2 \, \sqrt{a}}\right)^2 - \frac{b^2}{4 \, a}.
\end{align}
In this example the values of $(a,b)$ are $(1, 1)$. 
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