Question

2x + 2\(\sqrt{\left(x+5\right)x}\)+\(\sqrt{x+5}+\sqrt{x}=25\)

giải phương trình

Answer 1

ĐKXĐ: \(x\ge0\)

Đặt \(\sqrt{x}+\sqrt{x+5}=a>0\)

\(\Rightarrow a^2=2x+5+2\sqrt{x\left(x+5\right)}\)

Phương trình trở thành:

\(a^2-5+a=25\)

\(\Leftrightarrow a^2+a-30=0\Rightarrow\left[{}\begin{matrix}a=5\\a=-6\left(l\right)\end{matrix}\right.\)

\(\Rightarrow\sqrt{x}+\sqrt{x+5}=5\)

\(\Leftrightarrow\sqrt{x}-2+\sqrt{x+5}-3=0\)

\(\Leftrightarrow\frac{x-4}{\sqrt{x}+2}+\frac{x-4}{\sqrt{x+5}+3}=0\)

\(\Leftrightarrow\left(x-4\right)\left(\frac{1}{\sqrt{x}+2}+\frac{1}{\sqrt{x+5}+3}\right)=0\)

\(\Rightarrow x=4\)