Question

How do you solve `x^2-2x-4>0` using a sign chart?

Answer 1

The solution is `x in (-oo, (1-sqrt5)) uu((1+sqrt5), +oo)`

Explanation:

We must first find the roots of the quadratic equations

`x^2-2x-4=0`

The discriminant is

`Delta=b^2-4ac=(-2)^2-4*(1)*(-4)=20`

As `Delta>0`, there are `2` real roots

The roots are

`x=(-b+-sqrtDelta)/(2a)`

`x_1=(2-sqrt20)/2=1-sqrt5`

`x_2=(2+sqrt20)/2=1+sqrt5`

Let `f(x)=x^2-2x-4`

We can build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaaaa)` `-oo` `color(white)(aaaa)` `x_1` `color(white)(aaaa)` `x_2` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `(x-x_1)` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `(x-x_2)` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaaaa)` `+` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

Therefore,

`f(x)>0` when `x in (-oo, (1-sqrt5)) uu((1+sqrt5), +oo)`