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

1 Answer
Dec 24, 2016

Answer:

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[ #