Question

How do you solve `x^2-2x-5<=0` using a sign chart?

Answer 1

The answer is `1-sqrt6<=x<=1+sqrt6`

Explanation:

First we start by solving `y=x^2-2x-5=0`
We calculate `Delta=b^2-4ac=4+20=24`

So the roots are `=(2+-sqrt24)/2=(2+-2sqrt6)/2`
so the roots are `1+sqrt6` and `1-sqrt6`
We can do the sign chart

`x` `color(white)(aaaaaa)` `-oo` `color(white)(aaa)` `1-sqrt6` `color(white)(aaa)` `1+sqrt6` `color(white)(aaa)` `+oo`
`1-sqrt6` `color(white)(aaaa)` `-` `color(white)(aaa)` `0` `color(white)(aaaa)` `+` `color(white)(aaaaa)` `+`
`1+sqrt6` `color(white)(aaaa)` `-` `color(white)(aaa)` ###color(white)(aaaaa)#`-` `color(white)(aaaaa)` `+`
`y` `color(white)(aaaaaaaaa)` `+` `color(white)(aaaaaaaa)` `-` `color(white)(aaaaa)` `+` `color(white)(aaaaa)`

So `x^2-2x-5<=0` when `1-sqrt6<=x<=1+sqrt6`