Finding square roots of an unknown

Question

$p$ and $q$ are rational numbers, find the values of p and q given that $2p = q \sqrt {76} + \sqrt {19}$

attempt: $$2p = 2q \sqrt{19} + \sqrt{19}$$ $$2p = \sqrt{19}(2q+1) $$

How to continue or how should I do?

Answer 1

Observe that the left side is rational, but the right side will be irrational unless one of the numbers is zero. This comes from the fact that the product of an irrational number and a nonzero rational number is always irrational.

Thus one of the brackets on the RHS must be zero, which implies $2q + 1 =0$. So we must have $p = 0, q = -\frac{1}{2}$.