Much of one's mathematical experience prior to the calculus is spent
in solving equations. There are the kind of equations, known as
*identities*, where every number in the domain is a solution.
The equation

is such an identity. It is not this, but the other type of equation, known as a

Another fact about conditional equalities is that comparatively few
of them may be solved exactly. Leaps in technology have made it
commonplace for students, with the purchase of a handheld calculator,
to have at their fingertips powerful graphing capability and numerical
methods for finding approximate solutions to many, perhaps even
most conditional equalities. This does not mean that one should
forego learning the algebraic techniques which lead to exact solutions,
thinking that deftness in pushing the right pair of buttons is an
appropriate substitute for the thinking processes such algebraic
methods introduce. Still, there is added value in the knowledge one
gets by investigating graphical methods. By these methods
one comes to think of the solutions of, say,

as the -values of points of intersection between the graphs of the two functions

As another example, solutions of the equation

would be found at points of intersection between the graphs of

It is in solving equations like this latter one that students become confused. What some students do is the following:

In summary, the error occurs in finding the zeros of a function and taking to be the roots of the equation, when the two concepts do not coincide. There is a simple way to make them coincide. We simply make one side zero (using a valid algebraic step, of course).

- Solutions of an equation like correspond to points of intersection between the two sides, considered as functions, of the equation (i.e., the function and the function, in this case a constant one, ).
- If the two functions are combined into one function on one side of the equation, there is still a second function, the zero function, that remains on the other side. Now we have in place of the old problem a new one (but entirely equivalent) of finding the solutions that correspond to points of intersection between the new left-hand side (in this case ) and the new right-hand side (here zero).
- When our combining of terms has left one side of the equation zero (which, when considered as a function, has the -axis as its graph), one may solve the equation by finding the zeros of the nonzero side of the equation.

Top Algebra Errors Made by Calculus Students (full document)

Full List of Grading Codes

Thomas L. Scofield 2003-09-04