Is this what is meant by quadratic equation , function?

Question

It says in my textbook that :

Let $f(x) - g(x) = 0$. Then, let this equation be $h(x)$.
If $h(x)$ is a quadratic Q.F, then $h(x) = 0$ is a quadratic equation.

I need to ask some questions regarding this statement i.e.

$1)$ When I say $f(x)-g(x) = 0$. So, is this statement an equation? And $f(x) - g(x)$ is a function until there is no equal to sign?

$2)$ Why did we equate $f(x)-g(x)$ to $h(x)$ and then call $h(x)$ as a quadratic function?

$3)$ When I say $h(x) = 0$ is a quadratic equation, there is still a function ${ h(x) }$ which we wrote. Shouldn’t we write the value of $h(x)$ (Let it be for example $2x^2 +x+4$) and then call it a quadratic equation?

Answer 1

That is a strange phrasing to me.

I would say let $f(x)-g(x)$ be $h(x)$. If $h(x)$ is a quadratic function, then $h(x)=0$ is a quadratic equation.

1. Yes, it is an equation. $f(x)-g(x)$ is a function of $x$.

2. We don't just call it a quadratic function, we have to verify that.

3. When we mention quadratic equation, we should see the equal sign somewhere.

Answer 2

$1)$ $f(x)-g(x)=0$ is an equation. $ f(x)-g(x)$ is a function.

$2)$ $h(x)$ being equal to an equation makes no sense. If your book says instead that $h(x) =f(x)-g(x)$, then $h(x)$ is really no quadratic function. It is the zero function. The degree of a quadratic is $2$, whereas the degree of the zero function is undefined. I believe this answers $(3)$ as well.