Question

What is the vertex form of `y=(2x+1)(x +6) + 6x`?

Answer 1

`y=color(green)(2)(x-(color(red)(-19/4)))^2+(color(blue)(-313/8))`

with vertex at `(color(red)(-19/4),color(blue)(-313/8))`

Explanation:

Given
`color(white)("XXX")y=color(brown)((2x+1)(x+6))+6x`

(first converting into standard form):
`color(white)("XXX")rArr y=color(brown)(2x(x+6)+1(x+6))+6x`

`color(white)("XXX")rArr y =color(brown)( 2x^2+12x+x+6)+6x`

`color(white)("XXX")rArr y= 2x^2+19x+6`

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

Extract the `color(green)(m)` factor
`color(white)("XXX")y=color(green)(2)(x^2+19/2x)+6`

Complete the square
`color(white)("XXX")y=color(green)(2)(x^2+19/2x+(19/4)^2)+6-2*(19/4)^2`

Simplify into vertex form
`color(white)("XXX")y=color(green)(2)(x-(color(red)(-19/4)))^2+(color(blue)(-313/8))`

Since `-19/4` is about `-5`
and `-313/8` is about `-39`
the graph below of the original equation supports this answer.
graph{(2x+1)(x+6)+6x [-19.8, 12.24, -40.05, -24.01]}