Question

Решить уравнение подробно

Answer 1

$20*9^x-12^x-16^x=0\\\\20*\frac{9^x}{16^x}-\frac{12^x}{16^x}-\frac{16^x}{16^x}=0\\\\20(\frac{9}{16})^x-(\frac{12}{16})^x-1=0\\\\20(\frac{3}4)^{2x}-(\frac{3}4)^x-1=0\\\\(\frac{3}4)^x=t;t\in(0;+\infty)\\20t^2-t-1=0\\D=1+4*20=81\\t_1=\frac{1+9}{40}=\frac{1}4\\t_2=\frac{1-9}{40}=-\frac{1}5,t\in(0;+\infty)\\\\t=\frac{1}4\\(\frac{3}4)^x=\frac{1}4\\x=log_{\frac{3}4}\frac{1}4$

Answer 2

20*9^x-12^x-16^x=0 |: 16^x
20*(9/16)^x-(12/16)^x-(16/16)^x=0
20*(3/4)^2x-(3/4)^x-1=0
(3/4)^x=t, t>0
20t²-t-1=0
D=81
t₁=1/4, t₂=-1/5, -1/5∉(0;∞)
t=1/4
(3/4)^x=1/4
log₃/₄(3/4)^x=log₃/₄(1/4)
x=log₃/₄(1/4)

если бы уравнение имело вид:
2*9^x-12^x-16^x=0, 
то получилось бы:
2t²-t-1=0
D=9
t₁=1, t₂=-1/2. -1/2∉(0;∞)
t=1
(3/4)^x=1
(3/4)^x=(3/4)⁰
x=0