Question

Let $f\left(x\right)=|x^2-9|-|x-a|,$ then number of integers in the range of $a$ so that $f\left(x\right)=0$ has 4 distinct real roots, is

Answer 1

$|x^2-9|=|x-a|$

For tangent $\to x^2-9=x-a$
$\downarrow$
D = 0
$a=\frac{37}{4}$
Similarly $a=-\frac{37}{4}$ for left tangent
So for 4 distinct solution.
$a\in (\frac{-37}{4},-3)\cup (-3,3)\cup (3,\frac{37}{4})$
So total number of integers = 17.