Question

What is the vertex form of `y=3x^2-39x-90`?

Answer 1

`y=3(x-13/2)^2-867/4`
`color(white)("XXX")` with vertex at `(13/2,-867/4)`

Explanation:

The general vertex form is `y=color(green)m(x-color(red)a)^2+color(blue)b` with vertex at `(color(red)a,color(blue)b)`

Given:
`y=3x^2-39x-90`

extract the dispersion factor (`color(green)m`)
`y=color(green)3(x^2-13x) -90`

complete the square
`y=color(green)3(x^2-13xcolor(magenta)(+(13/2)^2)) -90 color(magenta)(-color(green)3 * (13/2)^2)`

re-writing the first term as a constant times a squared binomial
and evaluating `-90-3 *(13/2)^2` as `-867/4`

`y=color(green)3(x-color(red)(13/2))^2+color(blue)(""(-867/4))`

Answer 2

Vertex form of equation is `y= 3 (x - 6.5)^2-216.75`

Explanation:

`y= 3 x^2 -39 x -90` or

`y= 3 (x^2 -13 x) -90` or

`y= 3 (x^2 -13 x + 6.5^2)-3*6.5^2 -90` or

`y= 3 (x - 6.5)^2-126.75 -90` or

`y= 3 (x - 6.5)^2-216.75`

Vertex is `6.5, -216.75` and

Vertex form of equation is `y= 3 (x - 6.5)^2-216.75`

graph{3x^2-39x-90 [-640, 640, -320, 320]} [Ans]