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Bài 5:
a) Ta có: \(A=\left(\dfrac{4x+5\sqrt{x}-1}{x\sqrt{x}+2x-\sqrt{x}-2}-\dfrac{3\sqrt{x}+1}{x+\sqrt{x}-2}\right):\dfrac{x+4\sqrt{x}+4}{x-1}\)
\(=\left(\dfrac{4x+5\sqrt{x}-1}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}-\dfrac{\left(3\sqrt{x}+1\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\right):\dfrac{\left(\sqrt{x}+2\right)^2}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\)
\(=\dfrac{4x+5\sqrt{x}-1-3x-3\sqrt{x}-\sqrt{x}-1}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\cdot\dfrac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+2\right)^2}\)
\(=\dfrac{x+\sqrt{x}-2}{\left(\sqrt{x}+2\right)^3}\)
\(=\dfrac{\sqrt{x}-1}{x+4\sqrt{x}+4}\)
b) Ta có: \(A-1=\dfrac{\sqrt{x}-1-x-4\sqrt{x}-4}{x+4\sqrt{x}+4}\)
\(=\dfrac{-\left(x+3\sqrt{x}+5\right)}{x+4\sqrt{x}+4}\)
\(=\dfrac{-\left(x+2\cdot\sqrt{x}\cdot\dfrac{3}{2}+\dfrac{9}{4}\right)-\dfrac{11}{4}}{x+4\sqrt{x}+4}< 0\forall x\) thỏa mãn ĐKXĐ
nên A<1
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1: Ta có: \(\sqrt{x^2+3x+2}=1\)
\(\Leftrightarrow x^2+3x+2=1\)
\(\Leftrightarrow x^2+3x+1=0\)
\(\text{Δ}=3^2-4\cdot1\cdot1=5\)
Vì Δ>0 nên phương trình có hai nghiệm phân biệt là:
\(\left\{{}\begin{matrix}x_1=\dfrac{-3-\sqrt{5}}{2}\\x_2=\dfrac{-3+\sqrt{5}}{2}\end{matrix}\right.\)
![](https://rs.olm.vn/images/avt/0.png?1311)
bài 2
a) ĐKXĐ: a\(\ge\)0, a\(\ne\)1
b)P=\(\dfrac{1+\sqrt{a}-1+\sqrt{a}}{\left(1-\sqrt{a}\right)\left(1+\sqrt{a}\right)}\).\(\dfrac{1+\sqrt{a}}{\sqrt{a}}\)
P=\(\dfrac{2\sqrt{a}}{\left(1-\sqrt{a}\right)\left(1+\sqrt{a}\right)}.\dfrac{1+\sqrt{a}}{\sqrt{a}}\)
P=\(\dfrac{2}{1-\sqrt{a}}\)
c) thay a=4 vào biểu thức ta có
P=\(\dfrac{2}{1-\sqrt{4}}\)=\(\dfrac{2}{1-2}\)=-2
d) để P=9 thì
\(\dfrac{2}{1-\sqrt{a}}=9\)\(\Rightarrow\)2=9(1-\(\sqrt{a}\))
\(\Rightarrow\)2=9-\(9\sqrt{a}\)\(\Rightarrow\)\(9\sqrt{a}=7\)\(\Rightarrow\)\(\sqrt{a}=\dfrac{7}{9}\)
\(\Rightarrow a=\dfrac{49}{81}\)
bài 3
a) \(\sqrt{9x^2}=4\Rightarrow3x=4\)\(\Rightarrow\)\(x=\dfrac{4}{3}\)
b)\(\Rightarrow\)\(\left(x-\sqrt{5}\right)^2\)=0\(\Rightarrow x-\sqrt{5}=0\)
\(\Rightarrow x=\sqrt{5}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
- để phương trình có 2 nghiêm thì \(\Delta>0\)\(\Leftrightarrow\Delta=m^2+4.4.15\ge240\)nên phương trình có nghiệm với mọi m
- \(\Leftrightarrow\hept{\begin{cases}m\ne0\\\Delta^'=2^2+5m>0\end{cases}\Leftrightarrow m>-\frac{4}{5}}\)
![](https://rs.olm.vn/images/avt/0.png?1311)