Define $E(a,b,c) = a \cdot b^2 + c$. What value of $a$ is the solution to the equation $E(a,4,5) = E(a,6,7)$?
$E(a,4,5) = a \cdot 4^2 + 5 = 16a + 5$ and $E(a,6,7) = a \cdot 6^2 + 7 = 36a + 7.$ We set these equal to each other: $16a + 5 = 36a + 7.$ Now we simplify and have $20a=-2$, so $a = \boxed{-\frac{1}{10}}.$