Everything up to now has been focused on what we call singular
equations. Singular equations are simple in a particular way: they
are solved in a sequence of steps that unambiguously leads to a single
solution. One equation that is NOT singular is 
. It
has two solutions, x = 2 and x = 0. When solving it, we morph the
each side by taking its square root:
In doing this we get two possible values for the right hand side:
The techniques we've introduced so far all involve working with a
single picture; having a 'split' like this adds some complexity. Sometimes
we can adjust the situation in ways that let us continue with a single
picture (for example, if we work with the symbol 
). Yet there
are some cases, like what we have here, where that strategy will not
get you all the way to the final solutions. Other methods - conveniently
covered in this chapter! - can be employed to reveal the solutions
in these situations.
The definitions for singular and plural equations given before have been a little vague. This is because the concepts needed to give a precise definition had not been fully introduced. Now we can say that an equation is singular if can be solved by starting with a single equation image, and by applying symbol motions, fusions/fissions, and other morphs, arrive at a single solution. The key word is single. If we get a situation like the above, where we have two or more valid choices for where the equation is headed, it's a plural equation.
As we work the equation above, we arrive at a 'split': a point where a normal, single image cannot represent the two or more eventual solutions. As stated, at that point, the equations are
and
This chapter describes two strategies for solving plural equations internally. We'll introduce the first one by using it to finish solving this equation.