Question

What is the vertex form of `y=(x + 21)(x + 1)`?

Answer 1

`color(blue)("Vertex "-> (x,y)->(-11,-100)`

Explanation:

For a more detailed explanation of method see the example of
http://socratic.org/s/asZq2L8h .Different values but the method is sound.

Given:`" "y=(x+21)(x+1)`

Let `k` be the error correcting constant

Multiply out giving

`" "y=x^2+22x+21`

`y=(x^(color(magenta)(2))+22x)+21+k" "color(brown)("No error yet so k=0 at this stage")`

Move the power to outside the bracket

`y=(x+22color(green)( x))^(color(magenta)(2)) +21+k" "color(brown)("Now we have the error "->k!=0 )`

Remove the `x` from `22color(green)(x)`

`" "y=(x+color(red)(22))^2+21+k`

Multiply `color(red)(22)" by "(1/2) =color(blue)(11 )`

`" "color(green)(y=(x+color(red)(22))^2+21+k)`

`"changes to "color(green)(y=(x+color(blue)(11))^2+21+k)`

The error introduced is `(axxb/2)^2 ->(1xx22/2)^2 =+ 121`

So `k = -121` to 'get rid' of the error
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
So
`" "color (magenta)( y=(x+color(blue)(11))^2+21)-121`

`" "y=(x+11)^2-100`

`" "color(blue)("Vertex "-> (x,y)->(-11,-100)`