Question

What is the vertex form of `y=x^2 - 2 x - 5`?

Answer 1

Vertex form is `y= (x-1)^2 -6`

The vertex is the point `(1,-6)`

Explanation:

Vertex form is `y = a(x+p)^2 +q` with the vertex at `(-p, q)`

This is derived by the process of completing the square.

The quadratic trinomial `x^2 -2x-5` is in the form `ax^2 +bx+c`

`x^2 -2x-5` is not a perfect square.

We need to add the correct constant which will make it a square.
This is found from `(b/2)^2` which in this case is `((-2)/2)^2 = color(blue)(1)`

`y = x^2 -2x color(blue)(+1) color(blue)(-1) -5" "larr (color(blue)(+1-1=0))`

`y = (x^2 -2x color(blue)(+1))+ ( color(blue)(-1) -5)`

`y= (x-1)^2 -6" "larr` this is vertex form.

The vertex is the point `(1,-6)`