<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] Okay so $0.5 = \frac{1}{2}$ and $0.25 = \frac{1}{4}$.
Let's start from $$\left ( x + \frac{1}{2} \right )^{2} - \frac{1}{4}$$
$$= \left (\frac{2x+1}{2}  \right )^{2} - \frac{1}{4}$$
$$= \frac{(2x+1)^{2}}{4} - \frac{1}{4}$$
$$= \frac{(2x+1)^{2}-1}{4}$$
$$= \frac{4x^{2}+4x}{4}$$
Then factor out $4x$ in the numerator to obtain $$= \frac{4x(x+1)}{4}$$
Then you'll get $x^{2}+x$. 
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