Question

What is the vertex form of `y=2x^2+5x-3`?

Answer 1

`y=2(x-(-5/4))^2+(-6 1/8)`

Explanation:

General vertex form:
`color(white)("XXX")y=m(x-color(blue)(a))^2+color(orange)(b)` with vertex at `(color(blue)(a),color(orange)(b))`

Given
`color(white)("XXX")y=2x^2+5x-3`

Extract "spread factor" `m`
`color(white)("XXX")y=2(x^2+5/2x)-3`

Complete the square
`color(white)("XXX")y=color(red)(2)(x^2+5/2xcolor(red)(+(5/4)^2))-3-color(red)(2(5/4)^2)`

Write as a squared binomial and simplify the constant
`color(white)("XXX")y=2(x+5/4)^2-6 1/8`

Re-write to match signs of standard general form:
`color(white)("XXX")y=2(x-color(blue)(color(white)("")(-5/4)))^2+color(orange)(color(white)("")(-6 1/8))`