Question

How do you know if a polynomial of degree 2 has an x-intercept?

Answer 1

Once you have it in standard `ax^2+bx+c` form, check the discriminant `Delta = b^2-4ac` is non-negative.

Explanation:

Any polynomial of degree `2` in `x` can be expressed in the form:

`ax^2+bx+c`

This has discriminant `Delta` given by the formula:

`Delta = b^2-4ac`

If `Delta >= 0` then `ax^2+bx+c = 0` has at least one Real root.

Roots of `ax^2+bx+c = 0` are `x` intercepts.

The roots of `ax^2+bx+c = 0` are given by the formula:

`x = (-b+-sqrt(b^2-4ac))/(2a) = (-b+-sqrt(Delta))/(2a)`

Notice that the discriminant is the expression under the square root, so the square root takes Real values if and only if `Delta >= 0`.