Will a 2nd degree function always have maximum 2 roots?

Question

Will a function of 2nd degree always have maximum 2 roots?

For example: $$f(x) = x^2 - ln(x^2 +1) -1 $$

EDIT: More specific; if you have a function with $$k * x^2$$ where k is a real number, and this term is only added or subtracted to the function, not multiplied with any other term, does the function have maximum 2 roots?

Answer 1

More specific; if you have a function with $$k * x^2$$ where k is a real number, and this term is only added or subtracted to the function, not multiplied with any other term, does the function have maximum 2 roots?

No, take as an example $\sin x+0.01 x^2$, it has $6$ roots. Reducing the coefficient of the $x^2$ term will significantly increase the number of solutions. For instance, here is a plot of $\sin x+0.00001x^2$, see if you can count how many times it crosses the $x$-axis:

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Answer 2

Yes. And no.

A second order polynomial $p:\mathbb R\to\mathbb R$, i.e. a real function defined as

$$p(x) = ax^2+bx+c$$ where $a,b,c$ are real numbers, has a maximum of $2$ roots as it can have either $1$ root ($x^2$), $2$ roots ($x^2-1$) or $0$ roots ($x^2+1$).

A second order polynomial $p:\mathbb C\to\mathbb C$, i.e. a complex function defined as

$$p(x) = ax^2+bx+c$$ where $a,b,c$ are complex numbers, has $2$ roots or one double root.

In general, a function $f(x)$ in which $x^2$ appears may have as many roots as you want it to have. See $\sin(x^2)$ from the commentary in the example.