Question

How do you solve `n^2+2n-24>0`?

Answer 1

`-4< n>6`

Explanation:

`n^2+color(red)(2)ncolor(orange)(-24)> 0`

We need to factor this quadratic

We need to find two numbers that add to `2` and multiply to `-24`
`+2`
x`24`
..........
`color(black)(1)color(white)(.)color(black)(24)`
`color(black)(2)color(white)(.)color(black)(12)`
`color(black)(3)color(white)(.)color(black)(8)`
`color(green)(4)color(white)(.)color(green)(6)`

`4` and `6` can be combined to form `2`, but we'll need to subtract them.

We are looking for one positive number and one negative number. We can tell because `-24` can only be the product of one positive and negative number.

Now we just need to see if `-4+6` gives us `color(red)(2)` or if it's `-6+4`:
`-4+6=2`
`-6+4=-2`

So, our numbers are `-4` and `6`, which gives us `(n-4)(n+6)> 0`

We just set each factor to zero and solve:
Case 1
`n-4> 0`

`n>4`

Case 2
`n+6> 0`

`n>6`