DEFINITION
quadratic equation
Let $\,a\,$, $\,b\,$ and $\,c\,$ be real numbers, with $\,a\ne 0\,$.
A quadratic equation is an equation of the form:
$$ax^2 + bx + c = 0$$
Important notes about the definition:

A quadratic equation is, first and foremost, an equation.
It must have an ‘$\,=\,$’ sign.

When mathematicians say ‘an equation of the form ...’
they really
mean ‘an equation that can be put in the form ...’
by using the two primary equationsolving tools:
the Addition Property of Equality
and the Multiplication Property of Equality.

A quadratic equation must have an
[beautiful math coming... please be patient]
$\,x^2\,$ term.
This is what $\,a\ne 0\,$ tells us.

A quadratic equation is allowed (but not required) to have an $\,x\,$ term.
The coefficient $\,b\,$ might be zero, which means the $\,x\,$ term is gone.

A quadratic equation is allowed (but not required) to have a constant term.
(Recall that a constant term is just a number—no variables.)
The constant term, $\,c\,$, might be zero.
So, to check if an equation is a quadratic equation,
you want to make two passes through it (both sides):

Does it have an $\,x^2\,$ term appearing somewhere?
If not, then it's not a quadratic equation.
Note: it can have lots of $\,x^2\,$ terms!

The only other two term types that are allowed are $\,x\,$ terms and constants terms.
(For example: no $\,x^3\,$ terms, no variables inside square roots, no variables in denominators, and so on.)
So, sweep across the equation and look for anything other than $\,x\,$ terms and constant terms.
If you find any, then it's not a quadratic equation.
EXAMPLES:
In this exercise, you will practice identifying quadratic equations.
Question:
Is
$\,x^2 = x + 4\,$ a quadratic equation?
Solution:
Does it have an $\,x^2\,$ term?
Check!
Anything other than $\,x\,$ terms or constant terms?
Nope.
Check!
YES, it is a quadratic equation.
Question:
Is
$\,3x  4 = x + 1\,$ a quadratic equation?
Solution:
Does it have an $\,x^2\,$ term?
Nope.
So, it's not a quadratic equation.
Question:
Is
$\,x  2x^2 = 1 + x^5\,$ a quadratic equation?
Solution:
Does it have an $\,x^2\,$ term?
Check!
Anything other than $\,x\,$ terms or constant terms?
Oops.
Quadratic equations are not allowed to have an $\,x^5\,$ term.
So, it's not a quadratic equation.