trigonometry
posted by ludvic .
classify the graph of the equation 4y^2+5X+3y+7=0 as a circle, a parabola, an ellipse, or a hyperbola.

since there's y^2 but no x^2, it's a parabola, opening to the right:
5x = 4y^2  3y  7
x = 4/5 y^2  3/5 y  7/5
x = 4/5 (y^2  3/4 y)  7/5
x = 4/5 (y^2  3/4 y + (3/8)^2)  7/5  4/5 * (3/8)^2
x = 4/5 (y  3/8)^2  121/80
so the vertex is at (121/80,3/8)
http://www.wolframalpha.com/input/?i=plot+4y^2%2B5X%2B3y%2B7+%3D+0