Răspuns:
Explicație pas cu pas:
[tex]DVA:~~x-2>0,~deci~x>2.\\notam~lg(x-2)=a,~atunci~lg^{2}(x-2)^{2}=(lg(x-2)^{2})^{2}=(2*lg(x-2))^{2}=4*a^{2}\\[/tex]
Inlocuind obtinem: 4a²+a-5=0, ecuatie de gradul 2.
Δ=1²-4·4·(-5)=1+80=81. √81=9, deci
[tex]a_{1}=\dfrac{-1-9}{2*4} =\dfrac{-10}{8}=-\dfrac{5}{4} \\a_{2}=\dfrac{-1+9}{2*4} =\dfrac{8}{8}=1\\ Deci~lg(x-2)=-\dfrac{5}{4},~x-2=10^{-\frac{5}{4}},~x=2+\frac{1}{10^{\frac{5}{4}}}>2.\\ Sau~lg(x-2)=1,~x-2=10^{1},~deci~x=10+2=12.[/tex]
Ambele solutii sunt valabile.
