Question

How do you solve `x^2-1>=4x` using a sign chart?

Answer 1

The answer is `x in ] -oo,(2-sqrt5) ] uu [(2+sqrt5), +oo[`

Explanation:

Let's rewrite the equation

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

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

We need the values of `x`, when `f(x)=0`

that is, `x^2-4x-1=0`

Let's calculate `Delta=16-4*1*-1=20`

`Delta>0`. there are 2 real solutions

`x_1=(4+sqrt20)/2=(4+2sqrt5)/2=2+sqrt5`

`x_2=(4-sqrt20)/2=(4-2sqrt5)/2=2-sqrt5`

Now, we can do our sign chart

`color(white)(aaaa)` `x` `color(white)(aaaaaaaaa)` `-oo` `color(white)(aaaa)` `(2-sqrt5)` `color(white)(aaaa)` `(2+sqrt5)` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x-2+sqrt5` `color(white)(aaaaa)` `-` `color(white)(aaaaaaaaaa)` `+` `color(white)(aaaaaaaaa)` `+`

`color(white)(aaaa)` `x-2-sqrt5` `color(white)(aaaaa)` `-` `color(white)(aaaaaaaaaa)` `-` `color(white)(aaaaaaaaa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaaaaaaa)` `+` `color(white)(aaaaaaaaaa)` `-` `color(white)(aaaaaaaaa)` `+`

Therefore,

`f(x>=0)` when `x in ] -oo,(2-sqrt5) ] uu [(2+sqrt5), +oo[`

graph{x^2-4x-1 [-11.25, 11.25, -5.625, 5.625]}