Deriving optical- and acoustical branches

[Elekko]

On the book "Introduction to Solid State Physics" by Kittel, on page 98 he derived the roots for optical and acoustical branches for the equation:

$$M_1 M_2 \omega^4-2C(M_1+M_2)\omega^2+2C^2(1-cos(Ka))=0$$

where the roots are:

$$\omega^2=2C(\frac{1}{M_1}+\frac{1}{M_2})$$ and
$$\omega^2=\frac{\frac{1}{2}C}{M_1+M_2}K^2 a^2$$

I'm wondering how he actually found these roots since he skipped the details? He only mentions the trigonometric identity can be set to zero... how are the roots found?

 

[TSny]

Note that these are the roots only for the limiting case of $Ka << 1$. As Kittel states, in this case you can let $\cos(Ka) \approx 1-\frac{1}{2}K^2a^2$.

Make this approximation and note that you have a quadratic equation in $\omega^2$.