Find values for x and/or y on the graph of x^2-2y^2+9x+8y-276=0 for which there is a vertical tangent line

x^2-2y^2+9x+8y-276=0

x^2+9x - 2y^2+8y = 276
x^2+9x+81/4 -2(y^2-4y+4) + 2(4) = 276
(x + 9/2)^2 - 2(y-2)^2 = 1153/4

This is just an hyperbola with vertices at (-9/2±√1153/2,2).