Question

Giải phương trình : \(\sqrt{\dfrac{1}{x+3}}+\sqrt{\dfrac{5}{x+4}}=4\)

Answer 1

ĐKXĐ: \(x>-3\)

Đặt \(x+3=t>0\)

\(\sqrt{\dfrac{1}{t}}+\sqrt{\dfrac{5}{t+1}}=4\)

\(\Leftrightarrow\sqrt[]{\dfrac{1}{t}}-2+\sqrt[]{\dfrac{5}{t+1}}-2=0\)

\(\Leftrightarrow\dfrac{1-4t}{\sqrt[]{t}+2t}+\dfrac{1-4t}{\sqrt[]{5\left(t+1\right)}+2\left(t+1\right)}=0\)

\(\Leftrightarrow\left(1-4t\right)\left(\dfrac{1}{\sqrt[]{t}+2t}+\dfrac{1}{\sqrt[]{5\left(t+1\right)}+2\left(t+1\right)}\right)=0\)

\(\Leftrightarrow1-4t=0\Rightarrow t=\dfrac{1}{4}\)

\(\Rightarrow x=-\dfrac{11}{4}\)