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Is there a formal name for equations and inequalities containing parameters (known variables, coefficients)? Is it just "equations with parameters"? "general equations"?

Here are some examples:

  • $$\sqrt{4x - 3} \cdot ln(5x - a) = \sqrt{4x - 3} \cdot ln(6x + a) $$
  • $$ (5^{2x + 1} - 25^x - 20)(\sqrt{ax - 6} - \sqrt{a - 2x}) = 0 $$
  • $$ \dfrac{log_ax}{x^2 + (a - 4)x + 4 - 2a} \le 0$$

I want to learn how to solve them, but my problem I cannot find resources in English on how to do it. I'm not a native English speaker, so I suspect this is because I'm translating it wrong. The literal translation from Russian would be "equations with a parameter", and usually such problems are stated in the following way: "find all the solutions of the equation depending on the value of $a$" or "find all the values of $a$ for which the equation has two distinct roots".

For example, when I search for YouTube videos on how to solve such equations in Russian with the search query "уравнения с параметром" ("equations with a parameter"), I get tons of relevant videos of different sorts, but when I do it in English with the same search query "equations with a parameter", I get almost nothing relevant. It makes me think that I'm doing something wrong.

Here's what I've found so far on Wikipedia:

Equation

Usually, the unknowns are denoted by letters at the end of the alphabet, x, y, z, w, ...,2 while coefficients (parameters) are denoted by letters at the beginning, a, b, c, d, ... . For example, the general quadratic equation is usually written $ax^2 + bx + c = 0$. The process of finding the solutions, or, in case of parameters, expressing the unknowns in terms of the parameters, is called solving the equation.

Parameter

Mathematical functions have one or more arguments that are designated in the definition by variables. A function definition can also contain parameters, but unlike variables, parameters are not listed among the arguments that the function takes. When parameters are present, the definition actually defines a whole family of functions, one for every valid set of values of the parameters. For instance, one could define a general quadratic function by declaring $f(x) = ax^2 + bx + c$.

But it doesn't really answer my question.

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2 Answers 2

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$x$ would be called a variable. $\sqrt{4x - 3} \cdot ln(5x - a) = \sqrt{4x - 3} \cdot ln(6x + a)$ would be called an equation. There is no other common word in usage that would be universally understood. Thats the best I can do unless you can narrow down your question more.

If you're looking to find these kinds of problems in english, they would just be referred to as olympiad problems, but this is much broader than what you want.

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I believe they would typically be called equations with restrictions, such as restricted domain or restrictions on solutions

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