For-x-R-solve-the-equation-below-2-x-4-3-4-x-2-3-4-x-2-x-6-3-

Question Number 19595 by khamizan833@yahoo.com last updated on 13/Aug/17

For x \in R, solve the equation below!

{(2^{x} - 4)}^{3} + {(4^{x} - 2)}^{3} = {(4^{x} + 2^{x} - 6)}^{3}

Commented by khamizan833@yahoo.com last updated on 13/Aug/17

show your work, please .

Answered by Tinkutara last updated on 13/Aug/17

Let 2^{x} = y .

{(y - 4)}^{3} + {(y^{2} - 2)}^{3} + {(6 - y - y^{2})}^{3} = 0

It is of the form where a + b + c = 0 so

a^{3} + b^{3} + c^{3} = 3 a b c = 0 as given .

(y - 4) (y^{2} - 2) (6 - y - y^{2}) = 0

\Rightarrow y = 4, \pm \sqrt{2}, 2

But only positive values of y are

allowed . So 2^{x} = 4, \sqrt{2}, 2

\Rightarrow x = 2, \frac{1}{2}, 1

Commented by khamizan833@yahoo.com last updated on 13/Aug/17

thank you sir . god bless you .