Proving a function has real roots I am not interested in finding roots but interested in proving that the function has real roots.
Suppose a function $f(x) = x^2 - 1$
This function obviously has real roots.
$x = {-1, 1}$
How could I prove this without actually finding the roots? 
Trial and error could work, number theory even? (modulus etc?) Calculus, any methods?
Thanks!

One way is using the discriminant of the quadratic equation:
$$\sqrt{b^2-4ac}$$
If the value inside the square root is greater than 0, then there are two real roots
If it is equal to 0, there is one real root
If it is less than 0, it has imaginary roots
