This Should Be Simple, Right?

Algebra Level 4

p(x)=51x2+mx+cq(x)=3x2+bx+a\begin{aligned} p(x)&=&51x^2+mx+c \\ q(x)&=&3x^2+bx+a \end{aligned}

p(x)p(x) and q(x)q(x) are two quadratic polynomials with integer coefficients such that p(r)=q(r)=0p(r)=q(r)=0. If rr is an irrational number, what is ca\dfrac{c}{a}?

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