Find the value of $c$ such that $h(k(x))=0$ has equal roots, where $h(x)=5x+2$ and $k(x)=cx^2-x+2$

Question

First I rearranged the equation:

$$h(k(x))=5(cx^2-x+2)+2=0$$ $$5cx^2-5x+12=0$$

So the next step would be to set the discriminant equal to $0$:

$$25-240c=0$$ $$c=\frac{5}{48}$$

Would $\frac{5}{48}$ be the value of $c$?