Question

Cho \(P=\left(2-\frac{\sqrt{x}-1}{2\sqrt{x}-3}\right):\left(\frac{6\sqrt{x}+1}{2x-\sqrt{x}-3}+\frac{\sqrt{x}}{\sqrt{x}+1}\right)\)

a. Rút gọn p

b Tính giá trị của  P khi = \(\frac{3-2\sqrt{2}}{4}\)

c. So sánh P với \(\frac{3}{2}\)

Answer 1

a: \(P=\dfrac{4\sqrt{x}-6-\sqrt{x}+1}{2\sqrt{x}-3}:\left(\dfrac{6\sqrt{x}+1+2x-3\sqrt{x}}{\left(2\sqrt{x}-3\right)\left(\sqrt{x}+1\right)}\right)\)

\(=\dfrac{3\sqrt{x}-5}{2\sqrt{x}-3}\cdot\dfrac{\left(2\sqrt{x}-3\right)\left(\sqrt{x}+1\right)}{2x+3\sqrt{x}+1}\)

\(=\dfrac{3\sqrt{x}-5}{2\sqrt{x}+1}\)

b: Thay \(x=\dfrac{3-2\sqrt{2}}{4}\) vào P, ta được:

\(P=\left(3\cdot\dfrac{\sqrt{2}-1}{2}-5\right):\left(2\cdot\dfrac{\sqrt{2}-1}{2}+1\right)\)

\(=\dfrac{\left(\dfrac{3}{2}\sqrt{2}-\dfrac{3}{2}-5\right)}{\sqrt{2}}\)

\(=\dfrac{\dfrac{3}{2}\sqrt{2}-\dfrac{13}{2}}{\sqrt{2}}=\dfrac{6-13\sqrt{2}}{4}\)