Question

How do you solve `x^4+36>=13x^2` using a sign chart?

Answer 1

The answer is `x in ] -oo,-3 ] uu [ -2,2 ]uu [3, oo[`

Explanation:

We need

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

Let's factorise the expression

`x^4-13x^2+36= (x^2-4)(x^2-9)`

`=(x+2)(x-2)(x+3)(x-3)`

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

We can now do the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-3` `color(white)(aaaa)` `-2` `color(white)(aaaa)` `2` `color(white)(aaaa)` `3` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x+3` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+` `color(white)(aaa)` `+`

`color(white)(aaaa)` `x+2` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+` `color(white)(aaa)` `+`

`color(white)(aaaa)` `x-2` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaa)` `+`

`color(white)(aaaa)` `x-3` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaa)` `+`

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

Therefore,

`f(x>=0)`, when `x in ] -oo,-3 ] uu [ -2,2 ]uu [3, oo[`