TITLE: What is the difference between an indeterminate and variable?
QUESTION [9 upvotes]: From Wikipedia-

In mathematics, and particularly in formal algebra, an indeterminate is a symbol that is treated as a variable, but does not stand for anything else but itself and is used as a placeholder in objects such as polynomials and formal power series.

So, how it is different from variable which is also used as a placeholder? What is the use of the indeterminate?

REPLY [9 votes]: $x^2 + 3x + 3$ is a polynomial which has $x$ as indeterminate.  The $x$ doesn't stand for any particular number, it's just the indeterminate of the expression.  You can ask various questions about the polynomial (e.g. "does it factor?"), but you can't ask "what is $x$?".
On the other hand, you might try to solve the equation $x^2 + 3x + 3 = 7$.  Here $x$ is being used as a variable:  it stands for an unknown number which makes the equation true when substituted for $x$.