Question

3*16^x+2*81^x=5*36^x, ecuatie exponentiala.

Answer 1

3*2^4x+2*3^4x=5*2^3x  * 3^2x
fie 2^2x=4^x=u  3^2x=9^x=v

3u²+2v²=5uv ecuatie omogena ; impartim prin v²
3(u/v)²-5(u/v)+2=0
u/v=w
3w²-5w+2=0

w1,2= [5+-√(25-24)]/6

w1=(5-1)/6=4/6=2/3
(4/9)^x=2/3⇒ x=1/2
w2=(5+1)/6=6/6=1
(4/9)^x=1 ⇒x=0

solitii x=0 si x=1/2
verificare 3+2=5 adevarat
3*4+2*9=5*6 adevarat , exercitiul este bine rezolvat

Answer 2

$3*16^x+2*81^x-5*36^x=0$
impartim prin $16^{x}$
$3+2*( \frac{9}{4}) ^{2x} -5* ( \frac{9}{4})^x=0$
fie $( \frac{9}{4})^x=t,t\ \textgreater \ 0$
$2t^2-5t+3=0$
Δ=1
$t=1 \\ t= \frac{3}{2}$
revenim la substitutie si obtinem  $( \frac{9}{4} )^{x}=1 ;( \frac{9}{4}) ^{x}= \frac{3}{2}$
$x=0 \\ x=1/2 \\ S={0;1/2}$