Question

What is the vertex form of `y=9x^2 - 21x + 10`?

Answer 1

`y=9(x-7/6)^2+(-9/4)` with vertex at `(x,y)=(7/6,-9/4)`

Explanation:

General vertex form is
`color(white)("XXX")y=color(green)(m)(x-color(red)a)^2+color(blue)b`
where
`color(white)("XXX")color(green)m` is a measure of the parabolic "spread";
`color(white)("XXX")color(red)a` is the `x` coordinate of the vertex; and
`color(white)("XXX")color(blue)b` is the `y` coordinate of the vertex.

Given
`color(white)("XXX")y=9x^2-21x+10`

Extract the spread factor `color(green)m`
`color(white)("XXX")y=color(green)9(x^2-7/3x)+10`

Complete the square for the first term and subtract a corresponding amount from the second
`color(white)("XXX")y=color(green)9(x^2-7/3xcolor(magenta)(+(7/6)^2))+10color(magenta)(-9 * (7/6)^2)`

Rewrite as a squared binomial and simplify the constant
`color(white)("XXX")y=color(green)9(x-color(red)(7/6))^2+color(blue)((-9/4))`

For verification purposes, here is the graph of this function (with grid lines at `1/12` units; note: `7/6=1 2/12` and `-9/4=-2 3/12`)