I am about to pull my hair ...

solution = {roots}

{solutions} = {}

In this context, it is standard to say 'a solution' to mean 'a value of $x$ that makes the equation true.'
i.e. {solutions of $1 + \sqrt{x^2 + 4} - x - \sqrt{2x + 1} = 0$ } = {roots of $1 + \sqrt{x^2 + 4} - x - \sqrt{2x + 1}$}

Using the plain English definition of 'solution', sure, there is only one solution to the problem - the set written above.
(Even then, what you've written is incorrect: {solutions} = {{roots}}.)

it is incorrect to refer to more than one solution to an equation.

While what you're suggesting does seem like a more sensible way of using the word, it is standard practice to use it the way I've described. Asking google/a teacher/a professor will confirm this.

it's just that in math we use words precisely ?

As I've said, ask google/a teacher/a professor. 'Precisely' does not mean 'perfectly in line with the everyday English definition'.

this is the math forum, not the "everyday English forum"

in math you can speak loosely of "solutions" when in fact you mean "the solution".

www.google.co.uk/search?q=%27solutions+of+an+equation%27
Seriously, just look it up. I've even googled it for you...

You don't have roots of an equation, you have roots of a function.
The solutions of $f(x) = 0$ are precisely the roots of $f(x)$.

This is how we use the word in mathematics. Find me a single source that suggests this is false; I've provided plenty that suggest it is true.

You don't have roots of an equation, you have roots of a function.
The solutions of $f(x) = 0$ are precisely the roots of $f(x)$.

This is how we use the word in mathematics. Find me a single source that suggests this is false; I've provided plenty that suggest it is true.

It's definitely correct that functions can have 'zeros' and that equations can have multiple 'solutions'.

However, I've done some digging around - there doesn't seem to be a consensus as to whether equations or functions (or both) have 'roots'.

So in our discussion, 'solutions' is correct but 'roots' may also be correct.

It's definitely correct that functions can have 'zeros' and that equations can have multiple 'solutions'.

However, I've done some digging around - there doesn't seem to be a consensus as to whether equations or functions (or both) have 'roots'.

So in our discussion, 'solutions' is correct but 'roots' may also be correct.

solution to the equation


(alternative solution suggested by 13 1 20 8 42)

Oh how amusing it would be to turn the page in a C3 or C4 exam to see that question

solution to the equation


(alternative solution suggested by 13 1 20 8 42)