2y = x² - 2x - 5

Method 1:

Get it in the form

    y = a(x - h)² + k

where the vertex is (h,k) and the two neighbor
points are (h±1,k+a)

Take one-half of the coefficient of x, which is 
one-half of -2 or -1.  Then square -1 and get
+1 then add +1  to both sides

2y + 1 = x² - 2x + 1 - 5

Factor the first three terms on the right

2y + 1 = (x - 1)(x - 1) - 5

Write (x - 1)(x - 1) as (x - 1)²

2y + 1 = (x - 1)² - 5

Add -1 to both sides:

    2y = (x - 1)x² - 6  

Divide every term by 2:

    y = (x - 1)² - 3

Compare to the standard equation:

    y = a(x - h)² + k

and we see that a = , h = 1, k= -3

So the vertex is

(h,k) = (1,-3) 

That's all you wanted but you should take
this opportunity to learn to graph.

The two neighbor points
are (h±1,k+a) = (1±1, -3+) which are
(0,) and (2,)

We can draw the graph by plotting those three 
points first

 

Then sketch a parabola through those three 
points:

 

Another way just to find the vertex is to
memorize the vertex formula:

----------------------

The general equation y = ax² + bx + c

has vertex (h,k) where

h = 

k = the value of y when h is substituted for x

-----------------------

So for 

2y = x² - 2x - 5

Get it in general form  y = ax² + bx + c

Divide through every term by 2

y = 

Compare to the general form

y = ax² + bx + c

and we see that

a = , b = -1, c = 

h =  =  =  =  = 1

k = the value of y when 1 is substituted for x 

y = 

y =  =  =  =  = 

So k = -3

So the vertex by this method is also (h,k) = (1,-3)

Edwin