Question

What is the vertex form of `5y = -3x^2-2x+2`?

Answer 1

`y = -3/5(x + 1/3)^2 + 7/15`

Explanation:

Given: `5y = -3x^2 - 2x + 2`

Vertex form: `y = a(x-h)^2 + k`,

where the vertex is `(h, k)` and `a` is a constant.

Complete the square to find the vertex form.

First divide by `5` to make `y =`:

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

Factor out `-3/5` so we only have an `x^2`:

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

To complete the square we need to multiply `1/2 * 2/3 = 1/3`. We also need to subtract the squared term that we added:

`-3/5 (x+1/3)(x+1/3) = color(red)(-3/5)(x^2 + 2/3x " "color(red)( + 1/9))`

`y = -3/5(x + 1/3)^2 + 2/5 - (color(red)(-3/5 (1/3)^2))`

`y = -3/5(x + 1/3)^2 + 3/3*2/5 + 1/15`

`y = -3/5(x + 1/3)^2 + 7/15`