Question

This is a quadratic form problem, where an overarching variable is given to an entire term in order to simplify the factoring process. I'll type it out in the comments section, please help.

Answer 1

\[\sqrt{x+6} - 6\sqrt[4]{x+6} +8 = 0\]

Answer 2

\[(x+6)^{1/2}-6(x+6)^{1/4}+8=0\]
To make it easier to see how to deal with the problem, consider the substitution \(t=(x+6)^{1/4}\). Then you can write the above equation as
\[\left((x+6)^{1/4}\right)^2-6(x+6)^{1/4}+8=0\\
t^2-6t+8=0\]
This can be factored to get \(t=4\) and \(t=2\). However, these aren't your solutions because they're in terms of \(t\), and not \(x\), the given variable.
\[t=2~~\Rightarrow~~(x+6)^{1/4}=2~~\Rightarrow~~x=2^4-6=10\]
\[t=4~~\Rightarrow~~(x+6)^{1/4}=4~~\Rightarrow~~x=4^4-6=250\]