Unknown operator x or x^{2}

Question

I am trying to generate the following pgfplot:

Minimum working example

\documentclass[12pt,a4paper]{article}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{tikz}
\usepackage{pgfplots}

\author{Some dude's name}
\begin{document}
\begin{tikzpicture}
\begin{axis}
[
    xlabel={x}, 
    ylabel={y}, 
    grid, 
    ticks=none
]
\addplot[blue, no marks, smooth, domain=0:1, samples=50] {25-5x -y};

\end{axis}
\end{tikzpicture}
\end{document}

but it throws an error multiple times:

 ! Package PGF Math Error: Unknown operator `x' or `x^' (in '25-5x^{2} -y^{2}').
See the PGF Math package documentation for explanation.
Type H <return> for immediate help.
...
l.19 ...omain=0:1, samples=50] {25-5x^{2} -y^{2}};
This error message was generated by an \errmessage
command, so I can't give any explicit help.
Pretend that you're Hercule Poirot: Examine all clues,
and deduce the truth by order and method.

In my messages/Log section, there are about 70x of the above error.

The Function that I want to Plot:

Function details:

Answer 1

Firstly, you can't omit the multiplication operation * in pgfmath expressions (this is not a LaTeX math formula!).

Secondly, you should always use a compat level with pgfplots, otherwise the behavior will be buggy (you have the following warning:

 Package pgfplots Warning: running in backwards compatibility mode (unsuitable tick labels; missing features). Consider writing \pgfplotsset{compat=1.18} into your preamble.

Finally, you are drawing a 3d-plot, so you need to use addplot3 (or similar functions, see the manual).

This works:

\documentclass[12pt,a4paper]{article}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{tikz}
\usepackage{pgfplots}\pgfplotsset{compat=1.18}

\author{Some dude's name}
\begin{document}
\begin{tikzpicture}
\begin{axis}
[
    xlabel={x}, 
    ylabel={y}, 
    grid, 
    ticks=none
]
\addplot3[blue, no marks, smooth, domain=0:1, samples=50] {25-5*x*x -y*y};

\end{axis}
\end{tikzpicture}
\end{document}

Notice the nice aliasing pattern due to the mesh... probably you want surf instead of smooth here. See manual at pages 124 and following: https://texdoc.org/serve/pgfplots/0#section.4.6

Answer 2

This is not perfect, but you can see it as a starting point:

\documentclass[12pt,a4paper]{article}
\usepackage[utf8]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{tikz}
\usepackage{pgfplots}
\pgfplotsset{compat=1.9}

\author{Some dude's name}
\begin{document}
\begin{tikzpicture}[declare function = {
            Z(\x,\y) = 25 - \x^2 - \y^2;
        }]
\begin{axis}
[
    xlabel={x}, 
    ylabel={y}, 
    grid, 
    ticks=none,
    view={100}{20}, 
    enlargelimits,
    domain=0:5,
    samples=50, 
    restrict z to domain=0:30
    ]
\addplot3[surf] {Z(x,y)};

\end{axis}
\end{tikzpicture}
\end{document}