Question

Lg(2x+6)-2lg(2x-3)=1

Answer 1

 Lg(2x+6)-2lg(2x-3)=1
2x+6>0 =>2x>-6 => x>-3/2
2x-3>0 => 2x>3 => x> 3/2 
1=lg10

lg(2x+6)-2lg(2x-3)=lg10
lg(2x+6)=2lg(2x-3)+lg10
lg(2x+6)=lg(2x-3)^2+lg10

lg(a)+lg(b)=lg(a*b)

lg10+lg(2x-3)^2=lg(10(2x-3)^2)
lg(2x+6)=lg(10(2x-3)^2)
2x+6=10(2x-3)^2
10(2x-3)^2=2x+6
40x^2-120x+90=2x+6
40x^2-122x+84=0 
2(x-2)(20x-21)=0                se putea face si prin delta, dar prefer asa
x-2=0=>x=2
20x-21=0=>x=21/20 nu convine deoarece 21/20 < 3/2

R: x=2

Answer 2

lg(2x+6)-lg(2x-3)²=1,  2x+6>0 si 2x-3>0 =>x>-3 si x>3/2 => x>3/2
lg(2x+6)/(2x-3)²=1
(2x+6)/(2x-3)²=10
2x+6=10(4x²-12x+9)
x+3=20x²-60x+45
20x²-61x+42=0
Δ=3721-3360=361
x₁= (61+19)/40= 80/40= 2>3/2
x₂= (61-19)/40=42/40=21/20=10,5/10=1,05<1,5=3/2 deci nu e solutie

=> singura solutie este x=2

O seara buna!