Question

What is the vertex of `y= 5x^2-x-1+(2x-1)^2`?

Answer 1

vertex`=(5/18, -25/36)`

Explanation:

Start by expanding the brackets and simplifying the expression.

`y=5x^2-x-1+(2x-1)^2`
`y=5x^2-x-1+(4x^2-4x+1)`
`y=9x^2-5x`

Take your simplified equation and complete the square.

`y=9x^2-5x`
`y=9(x^2-5/9x+((5/9)/2)^2-((5/9)/2)^2)`
`y=9(x^2-5/9x+(5/18)^2-(5/18)^2)`
`y=9(x^2-5/9x+25/324-25/324)`
`y=9(x^2-5/9x+25/324)-(25/324*9)`
`y=9(x-5/18)^2-(25/color(red)cancelcolor(black)324^36*color(red)cancelcolor(black)9)`
`y=9(x-5/18)^2-25/36`

Recall that the general equation of a quadratic equation written in vertex form is:

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

where:
`h=`x-coordinate of the vertex
`k=`y-coordinate of the vertex

So in this case, the vertex is `(5/18,-25/36)`.