Question

What is the vertex form of `y=8x^2 − 6x + 128`?

Answer 1

`color(blue)(y_("vertex form")=8(x-3/8)^2+ 126 7/8`

`color(brown)(" explanation given in detail")`

Explanation:

Given: `" "y=8x^2-6x+128` ..........(1)

Write as `" "y=8(x^2-6/8x)+128`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(brown)("Now we start to change things a step at a time.")`

`color(green)("Change the bracket so that this part becomes:")`
`8{x-(1/2 xx6/8)}^2`

`color(green)("Now put back the constant giving:")`
`8{x-(1/2 xx6/8)}^2 +128`

`color(green)("But this change has introduced an error so we can not yet equate it")` `color(green)("to "y.)`

`y!= 8{x-(1/2 xx6/8)}^2 +128`

`color(green)("We fix that by adding another constant ( say k ) giving:")`

`y=8{x-(1/2 xx6/8)}^2 +128+k` .........................(2)

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(brown)("To find the value of "k)`

`color(green)("Equate (2) to (1) through "y)`

`8{x-(1/2 xx6/8)}^2 +128+k" " =" "8x^2-6x+128`

`8(x^2-3/8x-3/8x+9/64)+128+k" "=" "8x^2-6x+128`

`cancel(8x^2)-cancel(6x)+9/8+cancel(128)+k" "=" "cancel(8x^2)-cancel(6x)+cancel(128)`

`k=-9/8` ....................................(3)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Substitute (3) into (2)

`color(blue)(y_("vertex form")=8(x-3/8)^2+ 126 7/8`

Note`" " 3/8 = 0.375`

So
`color(blue)(" "x_("vertex") = (-1)xx(-3/8) = + 0.375)`
`color(blue)(" "y_("vertex")= 126 7/8 = 126.875`

Tont B