Terminology to describe that, eg, $ax^2+bx+c=0$ cannot be solved by simply moving terms around to isolate $x$

The following older (mostly before 1900) usage may be of interest. Equations of the form $ax^2 + c = 0$ such that $a \neq 0$ used to be called pure quadratic equations (sometimes incomplete quadratic equations) and equations of the form $ax^2 + bx + c = 0$ such that $a \neq 0$ and $b \neq 0$ used to be called affected quadratic equations (sometimes complete quadratic equations). See this google books search.

Footnote * on p. 280 of College Algebra by Edward A. Bowser:

1888 1st edition at google books and 1893 reprint of 1st edition at internet archive

The term adfected, or affected, was introduced by Vieta, about the year 1600, to distinguish equations which involve, or are affected with, different powers of the unknown quantity from those which contain one power only.

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