find the root : $x^8-24x^4+256x^2+144=0$

Question

find the root of $\mathbb{R} $:

$$x^8-24x^4+256x^2+144=0$$

my try:

$$t=x^2\\t^4-24t^2+256t+144=0\\t^4-(8×3)t^2+(8×2^5)t+(8×18)=0$$

now ?!!

thank you very much!

Answer 1

Note that $$x^8-24x^4+144=(x^4-12)^2$$

Answer 2

The equation can be factored as $$(x^2-8x+36)(x^2+8x+4)=0$$ So here are the roots: $$x_1 = -2(2+\sqrt{3})$$ $$x_2 = -2(2-\sqrt{3})$$ $$x_3 = 4-2\sqrt{5}i$$ $$x_4 = 4+2\sqrt{5}i$$