Question

`y=3x^2-12x+7` into vertex form?

Answer 1

`y = 3(x-2)^2-5`

Explanation:

When given the form:

`y = ax^2+bx+c`

The vertex form is:

`y = a(x-h)^2+k`

where `h = -b/(2a)` and `k = a(h)^2+b(h)+c`

Given: `y=3x^2-12x+7`

We observe that `a = 3` and substitute that value into the vertex form:

`y = 3(x-h)^2+k" [1]"`

Compute h:

`h = 12/(2(3))`

`h = 2`

Substitute into equation [1]:

`y = 3(x-2)^2+k" [1.1]"`

Compute k:

`k = 3(2)^2-12(2)+7`

`k =-5`

Substitute into equation [1.1]

`y = 3(x-2)^2-5`

Answer 2

`y=3(x-2)^2-5`

Explanation:

Vertex Form is `y=a(x-k)^2+h`
Expanding the above form:
`y=a(x^2-2kx+k^2)+h`
`y=ax^2-2akx+ak^2+h`
Comparing this form with the form given in the problem,
`y=3x^2-12x+7`
`y=(a)x^2-(2ak)x+(ak^2+h)`

We can see that
`a=3`
`-2ak=-12`
`ak^2+h=7`
From here we can solve for k and h.
`-2(3)k=-12`
`-6k=-12`
`k=2`
`3*2^2+h=7`
`12+h=7`
`h=-5`