integration - Regarding $x^2-a^2$ inside the argument of dirac delta

My undergraduate system textbook has this property in the appendix
$$\delta(x^2-a^2)=\frac{1}{2|a|}[\delta(x-a)+\delta(x+a)]$$
and I can't seem to derive the result

I tried the following:

$\int_{-\infty}^{\infty}f(x)\delta(x^2-a^2)\, dx$

let $u^2=x^2-a^2$ and $x=\sqrt{u^2-a^2}$ however I couldn't resolve the new limit of integral as $u$ must be strictly positive under this substitution. What is another way I can approach this?