We split the equations:
(1) \(\sqrt{4+\sqrt{8+4c}}=2\)
(2) \(\sqrt{2+\sqrt{2+c}}=2\sqrt{2}\)
Square both sides of both equations:
(1) \(4+\sqrt{8+4c}=4\)
(2) \(2+\sqrt{2+c}=8\)
Simplify and square both sides of both equations once again:
(1) \(8+4c=0\)
(2) \(2+c=36\)
We get:
\(c=-2\) and \(c=34\)
That is incorrect, because \(c\) needs to be the same value for the equations to work.
By commutative property, we get another two equations:
(3) \(\sqrt{4+\sqrt{8+4c}}=2\sqrt{2}\)
(4) \(\sqrt{2+\sqrt{2+c}}=2\)
Square both sides of both equations:
(3) \(4+\sqrt{8+4c}=8\)
(4) \(2+\sqrt{2+c}=4\)
Simplify and sqaure both sides of both equations once again:
(3) \(8+4c=16\)
(4) \(2+c=4\)
For both equations, we get \(\boxed{c=2}\)
Take that! Guest!
