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Bài 1.1
a. Để căn thức có nghĩa (CTCN) thì $2x-1\geq 0$
$\Leftrightarrow x\geq \frac{1}{2}$
b. Để CTCN thì $-2x+0,5\geq 0$
$\Leftrightarrow 0,5\geq 2x\Leftrightarrow x\leq \frac{1}{4}$
c. Để CTCN thì \(\left\{\begin{matrix} x-1\neq 0\\ \frac{1}{x-1}\geq 0\end{matrix}\right.\Leftrightarrow x-1>0\Leftrightarrow x>1\)
d. Để CTCN thì \(\left\{\begin{matrix} x^2+2021\neq 0\\ \frac{2022-x}{x^2+2021}\geq 0\end{matrix}\right.\Leftrightarrow 2022-x\geq 0\) (do $x^2+2021>0$ với mọi $x\in\mathbb{R}$)
$\Leftrightarrow x\leq 2022$
Bài 1.2
a. $3=\sqrt{9}>\sqrt{8}$
b. $-7=-\sqrt{49}> -\sqrt{51}$
c. $3+\sqrt{2}> 3+\sqrt{1}=4=2+2=2+\sqrt{4}> 2+\sqrt{3}$
d. $\sqrt{26}+3>\sqrt{25}+3=8=\sqrt{64}>\sqrt{63}$
e.
$\frac{1}{2}=\frac{2-1}{2}=\frac{\sqrt{4}-1}{2}> \frac{\sqrt{2}-1}{2}$
f.
Xét hiệu $5-2\sqrt{7}-(3-\sqrt{10})=2-(\sqrt{28}-\sqrt{10})$
$=2-\frac{18}{\sqrt{28}+\sqrt{10}}< 2-\frac{18}{\sqrt{2(28+10)}}$ (áp dụng BĐT $\sqrt{a}+\sqrt{b}\leq \sqrt{2(a+b)}$)
$=2-\frac{18}{\sqrt{76}}< 2-\frac{18}{\sqrt{81}}=0$
$\Rightarrow 5-2\sqrt{7}< 3-\sqrt{10}$
Câu 1:
1: Ta có: \(A=3\sqrt{25}-\sqrt{36}-\sqrt{64}\)
\(=3\cdot5-6-8\)
\(=15-6-8=1\)
Câu I:
2: Ta có: \(B=\dfrac{\sqrt{x}}{\sqrt{x}+1}+\dfrac{\sqrt{x}}{\sqrt{x}-1}-\dfrac{x+1}{x-1}\)
\(=\dfrac{\sqrt{x}\left(\sqrt{x}-1\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}+\dfrac{\sqrt{x}\left(\sqrt{x}+1\right)}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}-\dfrac{x+1}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\)
\(=\dfrac{x-\sqrt{x}+x+\sqrt{x}-x-1}{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\)
\(=\dfrac{x-1}{x-1}=1\)
Bài 8:
\(1,P=\dfrac{x+3\sqrt{x}+2+2x-4\sqrt{x}-2-5\sqrt{x}}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}=\dfrac{3x-6\sqrt{x}}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\\ P=\dfrac{3\sqrt{x}\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}=\dfrac{3\sqrt{x}}{\sqrt{x}+2}\\ 2,P=2\Leftrightarrow2\sqrt{x}+4=3\sqrt{x}\Leftrightarrow\sqrt{x}=4\\ \Leftrightarrow x=16\left(tm\right)\)
Bài 9:
\(a,M=\dfrac{x-1}{\sqrt{x}\left(\sqrt{x}-1\right)}:\dfrac{\sqrt{x}+1}{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}\\ M=\dfrac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\sqrt{x}\left(\sqrt{x}-1\right)}\cdot\left(\sqrt{x}-1\right)\\ M=\dfrac{x-1}{\sqrt{x}}\\ b,M>0\Leftrightarrow x-1>0\left(\sqrt{x}>0\right)\\ \Leftrightarrow x>1\)
Bài 10:
\(a,A=\dfrac{\sqrt{\left(x+3\right)^2}}{x+3}=\dfrac{\left|x+3\right|}{x+3}\)
Với \(x\ge-3\Leftrightarrow A=\dfrac{x+3}{x+3}=1\)
Với \(x< -3\Leftrightarrow A=\dfrac{-\left(x+3\right)}{x+3}=-1\)
\(b,B=\dfrac{2}{x-1}\cdot\dfrac{\left|x-1\right|}{2\left|x\right|}\)
Với \(0< x< 1\Leftrightarrow B=\dfrac{2}{x-1}\cdot\dfrac{-\left(x-1\right)}{2x}=-\dfrac{1}{x}\)
Bài 3:
Theo đề, ta có hệ phương trình:
\(\left\{{}\begin{matrix}\dfrac{1}{a}+\dfrac{1}{b}=\dfrac{1}{6}\\\dfrac{2}{a}+\dfrac{3}{b}=\dfrac{2}{5}\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\dfrac{3}{a}+\dfrac{3}{b}=\dfrac{1}{2}\\\dfrac{2}{a}+\dfrac{3}{b}=\dfrac{2}{5}\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}\dfrac{1}{a}=\dfrac{1}{10}\\\dfrac{1}{b}=\dfrac{1}{6}-\dfrac{1}{10}\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}a=10\\b=15\end{matrix}\right.\)
1.\(\Leftrightarrow\left\{{}\begin{matrix}4x+8y=28\\4x-5y=2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}4x=28-8y\\28-8y-5y=2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}4x=28-8y\\13y=26\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}4x=28-16\\y=2\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}x=3\\y=2\end{matrix}\right.\)
9.
a, \(x^4-x^3-14x^2+x+1=0\)
\(< =>x^4+3x^3-x^2-4x^3-12x^2+4x-x^2-3x+1=0\)
\(< =>x^2\left(x^2+3x-1\right)-4x\left(x^2+3x-1\right)-\left(x^2+3x-1\right)=0\)
\(< =>\left(x^2-4x-1\right)\left(x^2+3x-1\right)=0\)
\(=>\left[{}\begin{matrix}x^2-4x-1=0\left(1\right)\\x^2+3x-1=0\left(2\right)\end{matrix}\right.\)
giải pt(1) \(=>x^2-4x+4-5=0< =>\left(x-2\right)^2-\sqrt{5}^2=0\)
\(=>\left(x-2-\sqrt{5}\right)\left(x-2+\sqrt{5}\right)=0\)
\(=>\left[{}\begin{matrix}x=2+\sqrt{5}\\x=2-\sqrt{5}\end{matrix}\right.\)
giải pt(2) \(\)\(=>x^2+3x-1=0< =>x^2+2.\dfrac{3}{2}x+\dfrac{9}{4}-\dfrac{13}{4}=0\)
\(< =>\left(x+\dfrac{3}{2}\right)^2-\left(\dfrac{\sqrt{13}}{2}\right)^2=0\)
\(=>\left(x+\dfrac{3}{2}+\dfrac{\sqrt{13}}{2}\right)\left(x+\dfrac{3}{2}-\dfrac{\sqrt{13}}{2}\right)=0\)
tương tự cái pt(1) ra nghiệm rồi kết luận
b, đặt \(\sqrt{x^2+1}=a\left(a\ge1\right)=>x^2+1=a^2\)
\(=>x^4=\left(a^2-1\right)^2\)
\(=>pt\) \(\left(a^2-1\right)^2+a^2.a-1=0\)
\(=>a^4-2a^2+1+a^3-1=0\)
\(< =>a^4-2a^2+a^3=0< =>a^2\left(a+2\right)\left(a-1\right)=0\)
\(->\left[{}\begin{matrix}a=0\left(ktm\right)\\a=-2\left(ktm\right)\\a=1\left(tm\right)\end{matrix}\right.\)rồi thế a vào \(\sqrt{x^2+1}\)
\(=>x=0\)
8.31:
a: Xét ΔABD có AM/AB=AQ/AD
nên MQ//BD và MQ=BD/2
Xét ΔCBD có CN/CB=CP/CD
nên NP//BD và NP=BD/2
=>MQ//NP và MQ=NP
XétΔBAC có BM/BA=BN/BC
nên MN//AC
=>MN vuông góc BD
=>MN vuông góc MQ
Xét tứ giác MNPQ có
MQ//NP
MQ=NP
góc NMQ=90 độ
=>MNPQ là hình chữ nhật
=>M,N,P,Q cùng nằm trên 1 đường tròn
Câu 2:
Ta có: \(\sqrt{x^2-4x+4}=x-1\)
\(\Leftrightarrow2-x=x-1\left(x< 2\right)\)
\(\Leftrightarrow-2x=-3\)
hay \(x=\dfrac{3}{2}\left(tm\right)\)