Question

What is the vertex form of `y=2x^2+4x-5`?

Answer 1

`y=color(green)(2)(x-color(red)(""(-1)))^2+color(blue)(""(-8))`

Explanation:

Given:
`color(white)("XXX")y=2x^2+4x-5`

Remember that the vertex form is
`color(white)("XXX")y=color(green)(m)(x-color(red)(a))^2+color(blue)(b)`
with vertex at `(color(red)(a),color(blue)(b))`

Extracting the the `color(green)(m)` factor from the given equation
`color(white)("XXX")y=color(green)(2)(x^2+2x)-5`

Complete the square
`color(white)("XXX")y=color(green)(2)(x^2+2xcolor(purple)(+1))-5-color(green)(2)*color(purple)(1))`

Rewrite with a squared binomial and simplified constant
`color(white)("XXX")y=color(green)(2)(x-color(red)(""(-1)))^2+color(blue)(""(-8))`
which is the vertex form with vertex at `(color(red)(-1),color(blue)(-8))`

Here is a graph of the original equation for verification purposes:
graph{2x^2+4x-5 [-10.83, 11.67, -10.08, 1.17]}