Question

which function has zeros at $x = 10$ and $x = 2$?
$f(x) = x^2 - 12x + 20$
$f(x) = x^2 - 20x + 12$
$f(x) = 5x^2 + 40x + 60$
$f(x) = 5x^2 + 60x + 100$

Explanation:

Step1: Recall zero-to-function conversion

If a function has zeros at $x=a$ and $x=b$, the function can be written as $f(x) = k(x-a)(x-b)$ where $k
eq0$.

Step2: Substitute given zeros

Here, $a=10$ and $b=2$, so:
$$f(x) = k(x-10)(x-2)$$

Step3: Expand the expression

First multiply $(x-10)(x-2)$:
$$(x-10)(x-2) = x^2 -2x -10x +20 = x^2 -12x +20$$
For $k=1$, this matches the first option.

Step4: Verify (optional)

Substitute $x=10$: $f(10)=10^2-12(10)+20=100-120+20=0$
Substitute $x=2$: $f(2)=2^2-12(2)+20=4-24+20=0$

Answer:

A. $f(x) = x^2 -12x + 20$