Question

How do you solve `x^2+6x+9>=0` using a sign chart?

Answer 1

The answer is `x in RR`

Explanation:

We use

`(a+b)^2=a^2+2ab+b^2`

Factorise the expression

`x^2+6x+9=(x+3)^2`

Let `f(x)=(x+3)^2`

The domain of `f(x)` is `D_f(x)=RR`

and

`AA x in RR`, `f(x)>=0`

So,

`x in RR`

The sign chart is simple

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-3` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x+3` `color(white)(aaaa)` `-` `color(white)(aaa)` `0` `color(white)(aa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaa)` `+` `color(white)(aaa)` `0` `color(white)(aa)` `+`

Therefore,

`f(x)>=0` when `x in RR`