<s>[INST] Why does "$x^2 - 5x + 6 = 0$", which is the same as "$(x-3)(x-2) = 0$", represent a parabola? Consider the equation $x^2 - 5x + 6 = 0$. By factorising I get $(x-3)(x-2)  = 0$. Which means it represents a pair of straight lines, namely $x-2 =0 $ and $x- 3 = 0$, but when I plot $x^2 - 5x + 6 = 0$, I get a parabola, not a pair of straight lines. Why?
Plotting: x^2 - 5x + 6 = 0 
at Wolfram Alpha, I get the result:

 [/INST] 
I get $(x-3)(x-2) = 0$. which means it represent a pair of striaght
  lines namely $x-2 =0 $ and $x- 3 = 0$

When we just write
$$
(x-3)(x-2) = 0
$$
and ask for $x$ we usually mean the set of solutions
$$
S = \{ x \mid (x-3)(x-2) = 0 \}
$$
Two vertical lines in two dimensions we can model as:
$$
L_1 = \{ (x,y) \mid (x-3) = 0 \} \\
L_2 = \{ (x,y) \mid (x-2) = 0 \}
$$
These are different sets. We sloppily write $x-3=0$ and $x-2=0$ but mean the above.
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