<s>[INST] Can every parabola be written in the form of a quadratic $y=ax^2+bx+c$ or $x=dy^2+ey+f$? I understand that the graph of any equation of the form $y=ax^2+bx+c$ is a parabola (please correct me if I am mistaken).
My question is about the converse: Can every parabola be written in the form of an equation $y=ax^2+bx+c$ or $x=dy^2+ey+f$? 
 [/INST] A parabola is just the locus of points (a collection) that is equidistant from a point and a straight line. Let $(x,y)$ be the set of points, then for let's say you have a fixed point $(p,q)$ and a line $ax+by+c=0$, 
The distance between $(x,y)$ and the straight line is given by:
$$\frac{|ax+by+c|}{\sqrt{a^2+b^2}}$$
We want to set that to be equal to the distance between $(p,q)$ and $(x,y)$
$$ \frac{|ax+by+c|}{\sqrt{a^2+b^2}}=\sqrt{(x-p)^2+(y-q)^2}$$
$$\frac{a^2x^2+b^2y^2+2abxy+2acx+2bcy}{a^2+b^2}=x^2-2px+p^2+y^2-2qx+q^2$$

$$b^2x^2+a^2y^2-(2p(a^2+b^2)+2ac)x-(2q(a^2+b^2)+2bc)y-2abxy+(p^2+q^2)(a^2+b^2)=0$$
So the formula doesn't look pleasant, but we know from that, any parabola will be in the form of $ax^2+by^2+cx+dy+exy+f=0$
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