Question

How do you write `y=x^2-3x-2` in vertex form?

Answer 1

`y=(x-color(red)(3/2))^2+color(blue)(""(-17/4))` with vertex at `(color(red)(3/2),color(blue)(-17/4))`

Explanation:

Remember that that general 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))`

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

In this case we will ignore the `color(green)(m)` factor since it is equal to `color(green)(1)`

Completing the square
`color(white)("XXX")y=x^2-3xcolor(purple)(+(3/2)^2)-2 color(purple)(-(3/2)^2`

Rewriting as a squared binomial with a simplified constant
`color(white)("XXX")y=(x-color(red)(3/2))^2+(color(blue)(-17/4))`
which is the vertex form with vertex at `(color(red)(3/2),color(blue)(-17/4))`

Once more, here is the graph of the original equation for verification support:
graph{x^2-3x-2 [-3.524, 6.345, -4.77, 0.16]}