<s>[INST] Is it possible to find where two curves have a common value? Suppose I have;
$$f(x) = x^2 - 42x + 364$$
$$g(y) = y^2 - 35y + 364$$
Computing out values I find they have a common value 
$$f(6) = g(8) = g(27) = f(36) = 148$$
Is there a way of finding these values?

Working through based on received help...
(1) $$f(x) = g(y)$$
(2) $$f(x) - g(y) = x^2 - 42x + 364 - y^2 + 35y - 364 = 0$$
(3) $$x^2 - 42x - y^2 + 35y = 0$$
Complete the squares...
(4) $$(x^2 - 42x + 21^2) - (y^2 - 35y + \frac{35^2}{4}) = 21^2 - \frac{35^2}{4}$$
(5) $$4 ((x^2 - 42x + 21^2) - (y^2 - 35y + \frac{35^2}{4}) = 21^2 - \frac{35^2}{4} )$$
(6) $$(4x^2 - 168x + 42^2) - (4y^2 - 140y + 35^2) = 42^2 - 35^2$$
(7) $$(2x - 42)^2 - (2y - 35)^2 = 539$$
Considering (7) as Fermat's factorization method;
(8) $$a^2 - b^2 = c$$
(9) $$(a + b)(a - b) = 539$$
(10) $$(2x - 42 + 2y - 35)(2x - 42 - 2y + 35) = 539$$
(11) $$(2x + 2y - 77)(2x - 2y - 7) = 539$$
Then, considering (11) as a simple product of two terms gives;
(12) $$d * e = 539$$
(13) $$(2x - 2y - 7) = d$$
(14) $$(2x + 2y - 77) = 539/d$$
 [/INST] As the abovementioned 2 functions are continuous, you can find infinitely many common values in the intersection of their ranges.
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