If we temporarily replace x^2x2 with aa, we would have:
color(white)("XXX")a^2-12a+36XXXa2−12a+36
with fairly obvious factors (a-6)^2(a−6)2
Replacing the aa with the original x^2x2
we have
color(white)("XXX")x^4-12x^2+36=color(blue)((x^2-6)^2)XXXx4−12x2+36=(x2−6)2
If we regard 66 as (sqrt(6))^2(√6)2
then (x^2-6)(x2−6) can be thought of as the difference of squares with standard factors:
color(white)("XXX")(x^2-(sqrt(6)^2)=(x-sqrt(6))(x+sqrt(6))XXX(x2−(√62)=(x−√6)(x+√6)
and therefore the original equation can be further factored as
color(white)("XXX")x^4-12^x+36=[(x-sqrt(6))(x+sqrt(6))]^2XXXx4−12x+36=[(x−√6)(x+√6)]2
color(white)("XXXXXXXXXXX")=color(blue)((x-sqrt(6))^2(x+sqrt(6))^2)XXXXXXXXXXX=(x−√6)2(x+√6)2
