Question

What is the vertex form of `y=13x^2 +3x- 36`?

Answer 1

vertex form: `y=(x+3/26)^2-1881/52`

Explanation:

1. Factor 13 from the first two terms.

`y=13x^2+3x-36`

`y=13(x^2+3/13x)-36`

2. Turn the bracketed terms into a perfect square trinomial.
When a perfect square trinomial is in the form `ax^2+bx+c`, the `c` value is `(b/2)^2`. Thus you divide `3/13` by `2` and square the value.

`y=13(x^2+3/13x+(3/13x-:2)^2)-36`

`y=13(x^2+3/13x+9/676)-36`

3. Subtract 9/676 from the perfect square trinomial.
You cannot just add `9/676` to the equation, so you must subtract it from the `9/676` you just added.

`y=13(x^2+3/13x+9/676` `color(red)(-9/676))-36`

4. Multiply -9/676 by 13.
The next step is to bring `-9/676` out of the brackets. To do this, multiply `-9/676` by the `a` value, `13`.

`y=color(blue)13(x^2+3/13x+9/676)-36[color(red)((-9/676))*color(blue)((13))]`

5. Simplify.

`y=(x^2+3/13x+9/676)-36-9/52`

`y=(x^2+3/13x+9/676)-1881/52`

6. Factor the perfect square trinomial.
The last step is to factor the perfect square trinomial. This will allow you to determine the coordinates of the vertex.

`color(green)(y=(x+3/26)^2-1881/52)`

`:.`, the vertex form is `y=(x+3/26)^2-1881/52`.