Question

For what value of x^2-6x-7 negative?

Answer 1

`y <0` for `x in (-1,7)`

Explanation:

I assume you mean for what values of `x` is `x^2-6x-7` negative.

Let `y = x^2-6x-7`

So, we are looking for the values of `x` for which `y<0`

We can see that `y` is a quadratic function of the form `ax^2+bx+c` and since `a>0` `y` will have a minimum value.

Let's find the limiting cases where `y=0`

`:. x^2-6x-7=0`

`(x-7)(x+1) =0`

`:. x=+7 or -1`

Since `y` is a quadratic with a minimum value and `y=0` at `x=-1 and 7`, it holds that `y<0` between, but not equal to, these values of `x`.

We can write the above in interval notation as:

`y <0` for `x in (-1,7)`

We can see this more clearly by looking at the graph of `y` below.

graph{(x^2-6x-7) [-16.26, 24.3, -16.75, 3.51]}