Question

4 la puterea x + 2 la puterea x + 1 = 80/ ridic la puterea a 2-a
2 la puterea 2x + 2 la puterea x+1 = 80
Cum rezolv mai departe?

Answer 1

$Nicio~problema~:\\ \\ 4^x+2^{x+1}=80\\ \\ 2^{2x}+2*2^x=80.\\ \\ Notezi~astfel:~2^x=t,~t\in~N^{*}.\\ \\ t^2+2t-80=0\\ \\ \Delta=4+320=324=18^2.\\ \\ t_1=\frac{-2+18}{2}= 8\\ \\ t_2=\frac{-2-18}{2}=-10.\\ \\ Deoarece~2^x>0~\Leftrightarrow~t>0,~t_2~nu~convine!\\ \\ Solutia~unica~este~t_1=8~si~din~egalitatea~2^x=8\Rightarrow~x=3.$

Answer 2

4^x + 2^(x+1) = 80

(2^2)^x + 2^x ×2 = 80

2^x = a

a² + 2a = 80

a² + 2a - 80 = 0

Δ = b² - 4ac = 4 + 320 = 324

x1 = (-2 + 18)/2 = 16/2 = 8

x2 = (-2 - 18)/2 = -20 / 2 = -10 (nu este solutie )

2^x = 8

2^x = 2^3

x = 3