Question

\(\sqrt{x+2-4\sqrt{x-2}}+\sqrt{x+7-6\sqrt{x-2}}=1\)

Answer 1

ĐKXĐ: \(x\ge2\)

\(\sqrt{x-2-4\sqrt{x-2}+4}+\sqrt{x-2-6\sqrt{x-2}+9}=1\)

\(\Leftrightarrow\sqrt{\left(\sqrt{x-2}-2\right)^2}+\sqrt{\left(\sqrt{x-2}-3\right)^2}=1\)

\(\Leftrightarrow\left|\sqrt{x-2}-2\right|+\left|\sqrt{x-2}-3\right|=1\)

\(\Leftrightarrow\left|\sqrt{x-2}-2\right|+\left|3-\sqrt{x-2}\right|=1\)

Mặt khác ta có:

\(\left|\sqrt{x-2}-2\right|+\left|3-\sqrt{x-2}\right|\ge\left|\sqrt{x-2}-2+3-\sqrt{x-2}\right|=1\)

Dấu "=" xảy ra khi và chỉ khi:

\(2\le\sqrt{x-2}\le3\)

\(\Leftrightarrow6\le x\le11\)

Vậy nghiệm của pt là \(6\le x\le11\)