Question

How do you solve the inequality `2x^2-4x-1>0` and write your answer in interval notation?

Answer 1

The solution is `x in ]-oo,(1-sqrt6/2)[ uu ] (1+sqrt6/2), +oo[`

Explanation:

First, we must solve the equation

`2x^2-4x-1=0`

The discriminant is

`Delta=b^2-4ac=(-4)^2-4(2)(-1)=16+8=24`

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

`x=(-b+-sqrt(Delta))/(2a)`

`x_1=(4-sqrt24)/4=(4-2sqrt6)/4=1-sqrt6/2=-0.225`

`x_2=(4+sqrt24)/4=(4+2sqrt6)/4=1+sqrt6/2=2.225`

Let `f(x)=2x^2-4x-1`

Now, we build the sign chart

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

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

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

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

Therefore,

`f(x)>0`, when `x in ]-oo,(1-sqrt6/2)[ uu ] (1+sqrt6/2), +oo[`