Question

if 2x^2-3x+4y+5=0

find the vertex and the focus

Answer 1

2x^2-3x+4y+5=0
2x^2-3x+5 = -4y
x^2 - 3/2 x + 5/2 = -2y
x^2 - 3/2 x + (3/4)^2 = -2y + (3/4)^2 - 5/2
(x - 3/4)^2 = -2y - 31/16
(x - 3/4)^2 = -2(y + 31/32)

So, the vertex is at (3/4, -31/32)

You know that for the parabola

x^2 = 4py
here, 4p = -2, so p = -1/2

the focus is at a distance p from the vertex, so this parabola has a focus at (3/4, -31/32 + -1/2) = (3/4, -47/32)

Verify at
http://www.wolframalpha.com/input/?i=parabola+2x%5E2-3x%2B4y%2B5%3D0