Answer:
Algebraic "proof" that the solution is 2:
[tex]\sqrt{9-4\sqrt{5} }\:+\sqrt{5}[/tex]
[tex]=\sqrt{4-4\sqrt{5}+5 }\:+\sqrt{5}[/tex]
[tex]=\sqrt{2^2-4\sqrt{5}+(\sqrt{5})^2 }\:+\sqrt{5}[/tex]
[tex]=\sqrt{(2-\sqrt{5})^2 }\:+\sqrt{5}[/tex]
[tex]=(2-\sqrt{5})+\sqrt{5}[/tex]
[tex]=2[/tex]
However, this is not the correct mathematical solution to the problem.
The order of operations for [tex]\sqrt{(2-\sqrt{5})^2 }[/tex] dictate that the operation inside the parentheses must be carried out first.
As [tex]2-\sqrt{5} < 0[/tex], then [tex](2-\sqrt{5})^2[/tex] will always be positive.
If [tex](2-\sqrt{5})^2[/tex] is always positive, then [tex]\sqrt{(2-\sqrt{5})^2 }[/tex] will always be positive.
As √5 > 2 and [tex]\sqrt{(2-\sqrt{5})^2 } > 0[/tex] then [tex]\sqrt{(2-\sqrt{5})^2 }\:+\sqrt{5} > 2[/tex]
So mathematically, the actual solution to the expression is 2.47 (nearest hundredth) not 2.
