Question

Let f be a quadratic function such that

f(x) = ax +bx+c = a (x-h)² +k
If a > 0 and c < 0, how many real zeros will f(x) have?
02
04
3.
0 1
O 3
O none of the answer choices

Answer 1

The function will have 2 zeroes.

Explanation:

To find how many zeroes a quadratic function would have, we would have to look at its discriminant.

The discriminant (written using the capital greek letter Delta) of a quadratic function is defined like so:

`\text{if }y = ax^2 + bx + c\\\text{then }\Delta = b^2 - 4ac`

If the discriminant of a function is:
- Greater than 0, then it will have two zeroes.
- Equal to 0, then it will have one zero.
- Less than 0, then it will not have any zeroes.

Since we're given that
`a > 0` and
`c < 0`, then
`-4ac > 0`, as multiplying two negative numbers results in a positive number.

`b^2 \geq 0` for all b, as any number times itself is positive/zero (for the same reason that -4ac is positive).

Therefore,
`b^2 - 4ac > 0` for all a > 0 and c < 0.
Meaning that the function will have two zeroes.