UGAT Bangalore: AlgebraQuadratic Equations. Free, no login required.

The roots of the equation $\sqrt{2}x^2 - \frac{3}{\sqrt{2}}x + c = 0$ are $p$ and $2p$.

Let $a > 0$, and one root of the equation $a^2x^2 + 12a - 7 = 0$ is $64(p^6 + c^{12})$.

What is the value of $a$?