Question

Tìm x:

a) \(\sqrt{x+\sqrt{2x-1}}\) + \(\sqrt{x-2\sqrt{2x-1}}\) =\(\sqrt{2}\)

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

Answer 1

b) pt \(\Leftrightarrow\sqrt{2x+4+6\sqrt{2x-5}}+\sqrt{2x-4-2\sqrt{2x-5}}=4\)

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

Đk: \(x\ge\dfrac{5}{2}\)

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

TH1: \(\sqrt{2x-5}-1>0\Leftrightarrow x>3\)

(*) \(\Leftrightarrow\sqrt{2x-5}+3+\sqrt{2x-5}-1=4\Leftrightarrow2\sqrt{2x-5}=2\Leftrightarrow\sqrt{2x-5}=1\Leftrightarrow x=3\left(L\right)\)

TH2: \(\sqrt{2x-5}+3< 0\) (vô lý)

TH3: \(x\le3\)

(*) \(\Leftrightarrow\sqrt{2x-5}+3+1-\sqrt{2x-5}=4\Leftrightarrow4=4\) (luôn đúng)

KL: \(\dfrac{5}{2}\le x\le3\)

Answer 2

câu a, biểu thức trong dấu căn thứ 2 là \(x-2\sqrt{2x-1}\) hay \(x-\sqrt{2x-1}\) (có số 2 hay không?)