Weird Graph Behaviour - Analyzing a Root in Domain 2-4

[Mentallic]

I came across this weird graph from another thread:

y=\left(x^2-6x+8+\sqrt{x^4-12x^3+52x^2-96x+64}\right)^2

It seems that there is a root y=0 for the domain x=[2,4].
I've never seen such weird behaviour on a graph before. How could analysis of this function (or not?) determine that there is a root for all values 2\leq x\leq 4?

 

[Redbelly98]

That does seem weird, but I have figured out what is going on.

The expression under the radical is the square of (x²-6x+8). So that expression (including the square root) is simply the absolute value of (x²-6x+8).

They cancel whenever (x²-6x+8) is negative, which happens for 2<x<4.

 

[Mentallic]

Nice one Redbelly
I like this new little trick, it seems so nifty hehehe

Thanks.