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

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SUMMARY

The discussion focuses on the analysis of the function y=\left(x^2-6x+8+\sqrt{x^4-12x^3+52x^2-96x+64}\right)^2, revealing a root at y=0 for the domain x=[2,4]. The key insight is that the expression under the radical simplifies to the absolute value of (x^2-6x+8), which leads to cancellation when (x^2-6x+8) is negative, specifically within the interval 2 PREREQUISITES

  • Understanding of polynomial functions and their roots
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  • Knowledge of square roots and their properties
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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?
 
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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.
 
Nice one Redbelly :smile:
I like this new little trick, it seems so nifty hehehe

Thanks.
 

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