Write f(x) = 3x {}^(2) - 5x + 1 in the form of a(x - h)^(2) + k Help me please!​

Question

Write


`f(x) = 3x {}^(2) - 5x + 1`
in the form of

`a(x - h)^(2) + k`
Help me please!​

Answer 1

Work backwards: Expand the desired form to get

`a(x-h)^2+k=a(x^2-2xh+h^2)+k=ax^2-2ahx+ah^2+k`

Then
`a=3`,
`-2ah=-5`, and
`ah^2+k=1`, from which we get
`h=\frac56` and
`k=-(39)/(36)=-(13)/(12)`. So we end up with

`3x^2-5x+1=3\left(x-\frac56\right)^2-(13)/(12)`

Work forwards: Complete the square by writing

`3x^2-5x+1=3\left(x^2-\frac53x\right)+1=3\left(x^2-\frac53x+(25)/(36)-(25)/(36)\right)+1`

Now we have a perfect square trinomial:

`x^2-\frac53x+(25)/(36)=x^2-2\frac56x+\left(\frac56\right)^2=\left(x-\frac56\right)^2`

Finally,

`3x^2-5x+1=3\left(x-\frac56\right)^2-(13)/(12)`

same as before.