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a: \(P=\dfrac{x+\sqrt{x}+1+11\sqrt{x}-11+34}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}:\dfrac{x+\sqrt{x}+1-x+1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)
\(=\dfrac{x+12\sqrt{x}+24}{\sqrt{x}+2}\)
b: Thay \(x=3-2\sqrt{2}\) vào P, ta được:
\(P=\dfrac{3-2\sqrt{2}+12\left(\sqrt{2}-1\right)+24}{\sqrt{2}-1+2}\)
\(=\dfrac{27-2\sqrt{2}+12\sqrt{2}-12}{\sqrt{2}+1}=5+5\sqrt{2}\)
a. ĐK \(\hept{\begin{cases}x\ge0\\x\ne1\end{cases}}\)
b. M =\(\frac{\left(\sqrt{x}+1\right)^2-\left(\sqrt{x}-1\right)^2-5\sqrt{x}+1}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\)
\(=\frac{x+2\sqrt{x}+1-x+2\sqrt{x}-1-5\sqrt{x}+1}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}=\frac{1-\sqrt{x}}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\)
\(=\frac{-1}{\sqrt{x}+1}\)
c. \(M=\frac{-1}{\sqrt{x}+1}\ge-1\)
Vậy Min M =-1 khi x=0
Kết quả rút gọn: \(P=\frac{\sqrt{x}+2}{\sqrt{x}-1}\)
\(M=\frac{x+12}{\sqrt{x}-1}.\frac{\sqrt{x}-1}{\sqrt{x}+2}=\frac{x+12}{\sqrt{x}+2}\)
\(M=\frac{x-4+16}{\sqrt{x}+2}=\sqrt{x}-2+\frac{16}{\sqrt{x}+2}=\left(\sqrt{x}+2+\frac{16}{\sqrt{x}+2}\right)-4\)
Âp dụng BĐT AM-GM cho 2 số không âm ta có:
\(M\ge2\sqrt{\left(\sqrt{x}+2\right).\frac{16}{\sqrt{x}+2}}-4=2.4-4=4\)
Vậy min M =4. Dấu bằng xảy ra \(\Leftrightarrow\left(\sqrt{x}+2\right)^2=16\Leftrightarrow\sqrt{x}+2=4\Leftrightarrow\sqrt{x}=2\Leftrightarrow x=4\left(tm\right)\)
\(P=\left(\frac{3}{x-1}+\frac{1}{\sqrt{x}+1}\right):\frac{1}{\sqrt{x}+1}\) \(ĐKXĐ:x\ne1\)
\(P=\left(\frac{3}{x-1}+\frac{\sqrt{x}-1}{x-1}\right):\frac{1}{\sqrt{x}+1}\)
\(P=\frac{\sqrt{x}+2}{x-1}.\left(\sqrt{x}+1\right)\)
\(P=\frac{\sqrt{x}+2}{\sqrt{x}-1}\)
b) theo câu a) \(P=\frac{\sqrt{x}+2}{\sqrt{x}-1}\) với \(ĐKXĐ:x\ne1\)
theo bài ra \(P=\frac{5}{4}\)thì \(\Leftrightarrow\frac{\sqrt{x}+2}{\sqrt{x}-1}=\frac{5}{4}\)
\(\Leftrightarrow\left(\sqrt{x}+2\right).4=\left(\sqrt{x}-1\right).5\)
\(\Leftrightarrow4\sqrt{x}+8=5\sqrt{x}-5\)
\(\Leftrightarrow-\sqrt{x}+13=0\)
\(\Leftrightarrow-\sqrt{x}=-13\)
\(\Leftrightarrow\sqrt{x}=13\)
\(\Leftrightarrow x=169\)
vậy \(x=169\)khi \(P=\frac{5}{4}\)
