Question

\(\frac{1}{\sqrt{x-1}-\sqrt{x}}+\frac{1}{\sqrt{x-1}+\sqrt{x}}+\frac{\sqrt{x^3}-x}{\sqrt{x}-1}\)
Rút Gọn
Tính H khi x=\(\frac{53}{9-2\sqrt{3}}\)
Tính x khi H =16

Answer 1

Điều kiện xác định : \(0\le x\ne1\)

\(H=\frac{1}{\sqrt{x-1}-\sqrt{x}}+\frac{1}{\sqrt{x-1}+\sqrt{x}}+\frac{\sqrt{x^3}-x}{\sqrt{x}-1}\)

\(=\frac{\sqrt{x-1}+\sqrt{x}+\sqrt{x-1}-\sqrt{x}}{\left(x-1\right)-x}+\frac{x\left(\sqrt{x}-1\right)}{\sqrt{x}-1}\)

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

Với \(x=\frac{53}{9-2\sqrt{3}}\) tính H kết quả rất lẻ.H = 16 \(\Leftrightarrow\left(\sqrt{x-1}-1\right)^2=16\Leftrightarrow\left|\sqrt{x-1}-1\right|=4\)

\(\Leftrightarrow\sqrt{x-1}-1=4\) (Vì \(\sqrt{x-1}-1\ge-1>-4\))

\(\Leftrightarrow x=26\)