Question

What is the vertex form of `y=(3x+1)(x+2)+ 2`?

Answer 1

Vertex form is `y=3(x+7/6)^2-1/12` and vertex is `(-7/6,-1/12)`

Explanation:

Vertex form of quadratic equation is `y=a(x-h)^2+k`, with `(h,k)` as vertex.

To convert `y=(3x+1)(x+2)+2`, what we need is to expand and then convert part containing `x` into a complete square and leave remaining constant as `k`. The process is as shown below.

`y=(3x+1)(x+2)+2`

= `3x xx x+3x xx2+1xx x+1xx2+2`

= `3x^2+6x+x+2+2`

= `3x^2+7x+4`

= `3(x^2+7/3x)+4`

= `3(color(blue)(x^2)+2xxcolor(blue)x xxcolor(red)(7/6)+color(red)((7/6)^2))-3xx(7/6)^2+4`

= `3(x+7/6)^2-(cancel3xx49)/(cancel(36)^12)+4`

= `3(x+7/6)^2-49/12+48/12`

= `3(x+7/6)^2-1/12`

i.e. `y=3(x+7/6)^2-1/12` and vertex is `(-7/6,-1/12)`
graph{(3x+1)(x+2)+2 [-2.402, 0.098, -0.54, 0.71]}