Question

[Nguyễn Minh Toàn]

giải pt

\(\sqrt{x-7}+\sqrt{9-x}=x^2-16x+66\)

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

Answer 1

[Thanh Tùng DZ]

a) ĐKXĐ : \(7\le x\le9\)

đặt \(A=\sqrt{x-7}+\sqrt{9-x}\)

\(\Rightarrow A^2=2+2\sqrt{\left(x-7\right)\left(9-x\right)}\le2+\left(x-7\right)+\left(9-x\right)=4\)

\(\Rightarrow A\le2\)

Mà \(x^2-16x+66=\left(x-8\right)^2+2\ge2\)

\(\Rightarrow VT=VP=2\)

do đó : \(x-7=9-x\Leftrightarrow x=8\)( t/m )

b) ĐKXĐ : \(x\le1\)

Ta có : \(\sqrt{1-x}+\sqrt{\left(x-1\right)\left(x-2\right)}-\left|x-2\right|\sqrt{\frac{x-1}{x-2}}=3\)

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

\(\Leftrightarrow\sqrt{1-x}=3\Leftrightarrow x=-8\left(tm\right)\)