Question

Tìm tất cả các số x thoả: \(\sqrt{x-4\sqrt{x-2}+2}+\sqrt{x+6\sqrt{x-2}+7}=7\)

Answer 1

ĐKXĐ: \(x\ge2\)

\(\Leftrightarrow\sqrt{x-2-4\sqrt{x-2}+4}+\sqrt{x-2+6\sqrt{x-2}+9}=7\)

\(\Leftrightarrow\sqrt{\left(\sqrt{x-2}-2\right)^2}+\sqrt{\left(\sqrt{x-2}+3\right)^2}=7\)

\(\Leftrightarrow\left|\sqrt{x-2}-2\right|+\sqrt{x-2}+3=7\)

TH1: Nếu \(\sqrt{x-2}-2\ge0\Rightarrow x\ge6\)

\(\Rightarrow\sqrt{x-2}-2+\sqrt{x-2}+3=7\)

\(\Leftrightarrow2\sqrt{x-2}=6\Rightarrow\sqrt{x-2}=3\)

\(\Rightarrow x=11\) (t/m)

TH2: \(2\le x< 6\)

\(\Rightarrow2-\sqrt{x-2}+\sqrt{x-2}+3=7\)

\(\Leftrightarrow5=7\) (vô lý)

Vậy pt có nghiệm duy nhất \(x=11\)