\begin{array}{*{20}c} {x = \frac{{ - b \pm \sqrt {b^2 - 4ac} }}{{2a}}} & {{\rm{when}}} & {ax^2 + bx + c = 0} \\ \end{array}
This is all array stuff that you do not need. It has been used as a way to separate the three parts, that is foumula, when and initial equation
the 2 backslashes before it ate only a line break so they are definitely not needed.
\begin{array}{*{20}c} all the ampersands & and \end{array}
So I'm going to see what is left
x = \frac{{ - b \pm \sqrt {b^2 - 4ac} }}{{2a}}} & {{\rm{when} ax^2 + bx + c = 0
I don't know what the \rm{ } does. I looks like it is meant to be short for remark but i use \mbox{ } for that
\mbox is short for message box.
If we get rid of all that then this is what I have - the output is underneath
x = \frac{{ - b \pm \sqrt {b^2 - 4ac} }}{2a}}} when ax^2 + bx + c = 0
$$x = \frac{{ - b \pm \sqrt {b^2 - 4ac} }}{2a}}} when ax^2 + bx + c = 0$$
Now I want a space between the three sections so I am going to use the function \qquad (quad means 4 so I think this means leave 8 spaces. I could use \quad for a smaller gap
x = \frac{{ - b \pm \sqrt {b^2 - 4ac} }}{2a}}} \qquad when\qquad ax^2 + bx + c = 0
$$x = \frac{{ - b \pm \sqrt {b^2 - 4ac} }}{2a}}} \qquad when\qquad ax^2 + bx + c = 0$$
There you go - that is looking better.
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Now, for my own sake I want to look at this bit
\begin{array}{*{20}c} all the ampersands & and \end{array}
in particular this bit {*{20}c} it goes with the ampersands
It means something like
Put the formula at the begining&use the next 20 spaces for when¢re the last bit
I am not really sure what it does and it would be really good if HEUREKA tells us.
He know everything about LaTex. 
