Question

How do you solve `6n ^ { 2} + 11n - 65= 0`?

Answer 1

`n = 5/2" "` or `" "n = -13/3`

Explanation:

Given:

`6n^2+11n-65 = 0`

Let's attempt to use an AC method:

Look for a pair of factors of `AC=6 * 65 = 2 * 3 * 5 * 13 = 390` which differ by `11`

Note that `390` is close to `400 = 20^2`, so the numbers we are looking for are roughly `20+-11/2`.

We find the pair `26, 15` works in that `26 * 15 = 390` and `26 - 15 = 11`

Use this pair to split the middle term and factor by grouping:

`0 = 6n^2+11n-65`

`color(white)(0) = 6n^2+26n-15n-65`

`color(white)(0) = (6n^2+26n)-(15n+65)`

`color(white)(0) = 2n(3n+13)-5(3n+13)`

`color(white)(0) = (2n-5)(3n+13)`

Hence:

`n = 5/2" "` or `" "n = -13/3`