1.Tính M=\(\frac{\sqrt{28-6\sqrt{3}}-1}{28-6\sqrt{3}+\sqrt{28-6\sqrt{3}+1}}\)
2. Tìm x ϵ Z sao cho C ϵ Z biết:
C=\(\frac{\sqrt{x}}{x+\sqrt{x}+1}\)
Cho biểu thức sau: Q= \(\frac{-5\sqrt{x}+4}{3\sqrt{x}-2}+\frac{6\sqrt{x}+4}{2\sqrt{x}+3}+\frac{29\sqrt{x}-28}{3\left(6x+5\sqrt{x}-6\right)}\)
1) tìm điều kiện để Q có nghĩa và rút gọn Q
2) tìm x sao cho Q > \(\frac{8}{3}\)
3) Tìm x\(\in\) Z sao cho \(Q\in Z\)
1) ĐKXĐ: \(\left\{{}\begin{matrix}x\ge0\\x\ne\frac{4}{9}\end{matrix}\right.\)
Ta có: \(Q=\frac{-5\sqrt{x}+4}{3\sqrt{x}-2}+\frac{6\sqrt{x}+4}{2\sqrt{x}+3}+\frac{29\sqrt{x}-28}{3\left(6x+5\sqrt{x}-6\right)}\)
\(=\frac{3\left(-5\sqrt{x}+4\right)\left(2\sqrt{x}+3\right)}{3\left(3\sqrt{x}-2\right)\left(2\sqrt{x}+3\right)}+\frac{3\left(6\sqrt{x}+4\right)\left(3\sqrt{x}-2\right)}{3\left(2\sqrt{x}+3\right)\left(3\sqrt{x}-2\right)}+\frac{29\sqrt{x}-28}{3\left(2\sqrt{x}+3\right)\left(3\sqrt{x}-2\right)}\)
\(=\frac{3\left(-10x-7\sqrt{x}+12\right)}{3\left(3\sqrt{x}-2\right)\left(2\sqrt{x}+3\right)}+\frac{3\left(18x-8\right)}{3\left(2\sqrt{x}+3\right)\left(3\sqrt{x}-2\right)}+\frac{29\sqrt{x}-28}{3\left(2\sqrt{x}+3\right)\left(3\sqrt{x}-2\right)}\)
\(=\frac{-30x-21\sqrt{x}+36+54x-24+29\sqrt{x}-28}{3\left(2\sqrt{x}+3\right)\left(3\sqrt{x}-2\right)}\)
\(=\frac{24x+8\sqrt{x}-16}{3\left(2\sqrt{x}+3\right)\left(3\sqrt{x}-2\right)}\)
\(=\frac{8\left(3x+3\sqrt{x}-2\sqrt{x}-2\right)}{3\left(2\sqrt{x}+3\right)\left(3\sqrt{x}-2\right)}\)
\(=\frac{8\left(\sqrt{x}+1\right)\left(3\sqrt{x}-2\right)}{3\left(2\sqrt{x}+3\right)\left(3\sqrt{x}-2\right)}\)
\(=\frac{8\sqrt{x}+8}{6\sqrt{x}+9}\)
2) Để \(Q>\frac{8}{3}\) thì \(Q-\frac{8}{3}>0\)
\(\Leftrightarrow\frac{8\sqrt{x}+8}{6\sqrt{x}+9}-\frac{8}{3}>0\)
\(\Leftrightarrow\frac{24\sqrt{x}+24}{3\left(6\sqrt{x}+9\right)}-\frac{8\left(6\sqrt{x}+9\right)}{3\left(6\sqrt{x}+9\right)}>0\)
\(\Leftrightarrow\frac{24\sqrt{x}+24-48\sqrt{x}-72}{9\left(2\sqrt{x}+3\right)}>0\)
mà \(9\left(2\sqrt{x}+3\right)>0\forall x\) thỏa mãn ĐKXĐ
nên \(-24\sqrt{x}-48>0\)
\(\Leftrightarrow-24\left(\sqrt{x}+2\right)>0\)
\(\Leftrightarrow\sqrt{x}+2< 0\)(Vô lý)
Vậy: Không có giá trị nào của x thỏa mãn \(Q>\frac{8}{3}\)
Cho biểu thức
P= \(\frac{\sqrt{x}\left(1-x\right)^2}{1+\sqrt{x}}\): [(\(\left[\left(\frac{1-x\sqrt{x}}{1-\sqrt{x}}\right)+\sqrt{x}.\left(\frac{1+x\sqrt{x}}{1+\sqrt{x}}\right)-\sqrt{x}\right]\)
a) RG P
b) Tìm các giá trị lớn nhất của x để P < 1
c) Tìm x ϵ Z để P ϵ Z
\(\dfrac{2\sqrt{x}-9}{x-5\sqrt{x}+6}-\dfrac{\sqrt{x}+3}{\sqrt{x}-2}-\dfrac{2\sqrt{x}+1}{3-\sqrt{x}}\) ......=A(ghi ngược xíu)
a) Tìm đkxđ của A
b)Rút gọn A
c)Tìm x ϵ Z để A ϵ Z
Tìm x,y,z biết:\(\sqrt{x-1}+\sqrt{y-2}+\sqrt{z-3}=6-\frac{1}{\sqrt{x-1}}-\frac{1}{\sqrt{y-2}}-\frac{1}{\sqrt{z-3}}\)
\(ĐK:x\ge1,y\ge2,z\ge3\)
\(PT\Leftrightarrow\sqrt{x-1}+\frac{1}{\sqrt{x-1}}+\sqrt{y-2}+\frac{1}{\sqrt{y-2}}+\sqrt{z-3}+\frac{1}{\sqrt{z-3}}=6\)
Theo bđt AM-GM thì \(VT\ge6\)
Dấu "=" xảy ra khi \(\hept{\begin{cases}\sqrt{x-1}=\frac{1}{\sqrt{x-1}}=1\\\sqrt{y-2}=\frac{1}{\sqrt{y-2}}=1\\\sqrt{z-3}=\frac{1}{\sqrt{z-3}}=1\end{cases}}\)\(\Rightarrow\hept{\begin{cases}x=1\\y=3\\z=4\end{cases}}\)
6.A=\(\left(\dfrac{2}{\sqrt{x}-3}+\dfrac{2\sqrt{x}}{x-4\sqrt{x}+3}\right):\dfrac{2\left(x-2\sqrt{x}+1\right)}{\sqrt{x}-1}\)
a) Rút gọn A
b)Tìm a ϵ Z để biểu thức A nhận giá trị nguyên
a) Ta có: \(A=\left(\dfrac{2}{\sqrt{x}-3}+\dfrac{2\sqrt{x}}{x-4\sqrt{x}+3}\right):\dfrac{2\left(x-2\sqrt{x}+1\right)}{\sqrt{x}-1}\)
\(=\dfrac{2\left(\sqrt{x}-1\right)+2\sqrt{x}}{\left(\sqrt{x}-3\right)\left(\sqrt{x}-1\right)}:\dfrac{2\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}-1\right)}\)
\(=\dfrac{4\sqrt{x}-2}{\left(\sqrt{x}-3\right)\left(\sqrt{x}-1\right)}\cdot\dfrac{1}{2\left(\sqrt{x}-1\right)}\)
\(=\dfrac{2\sqrt{x}-1}{\left(\sqrt{x}-3\right)\left(\sqrt{x}-1\right)^2}\)
\(C=\left(1-\frac{1}{\sqrt{x}+2}\right):\left(\frac{4-x}{x-\sqrt{x}-6}-\frac{\sqrt{x}-2}{3-\sqrt{x}}-\frac{\sqrt{x}-3}{\sqrt{x}+2}\right)\)
a) Tìm x để C>0
b) Tìm x thuộc Z để C thuộc Z
Rút gọn
a, \(\left(3+\sqrt{5}\right)\sqrt{14-6\sqrt{5}}\)
b, \(\frac{2}{3+\sqrt{7}}\) + \(\frac{\sqrt{28}}{2}\) - 2
c, \(\frac{3}{3\sqrt{7}-6}-\frac{3}{3\sqrt{7}+6}\)
d, \(\left(\sqrt{3}+1\right)\sqrt{\frac{14-6\sqrt{3}}{5+\sqrt{3}}}\)
e, \(M=\frac{\sqrt{x}-1}{\sqrt{x}+3}-\frac{7\sqrt{x}-3}{9-x}-\frac{3\sqrt{x}-x}{\sqrt{x}-3}\)
a)= \(\left(3+\sqrt{5}\right)\left(\sqrt{\left(3-\sqrt{5}\right)^2}\right)\)=\(\left(3+\sqrt{5}\right)\left(3-\sqrt{5}\right)=9-5=4\)
b)= \(\frac{2\left(3-\sqrt{7}\right)}{\left(3+\sqrt{7}\right)\left(3-\sqrt{7}\right)}+\frac{\sqrt{2^2.7}}{2}-2\)=\(\frac{2\left(3-\sqrt{7}\right)}{9-7}+\sqrt{7}-2\)=1
c) =\(\frac{3}{3\left(\sqrt{7}-2\right)}-\frac{3}{3\left(\sqrt{7}+2\right)}\)=\(\frac{1}{\sqrt{7}-2}-\frac{1}{\sqrt{7}+2}=\frac{\sqrt{7}+2-\left(\sqrt{7}-2\right)}{\left(\sqrt{7}+2\right)\left(\sqrt{7}-2\right)}\)=\(\frac{4}{7-4}=\frac{4}{3}\)
d) =\(\left(\sqrt{3}+1\right)\sqrt{\frac{\left(14-6\sqrt{3}\right)^{ }\left(5-\sqrt{3}\right)}{\left(5+\sqrt{3}\right)\left(5-\sqrt{3}\right)}}\)=\(\left(1+\sqrt{3}\right)\sqrt{\frac{\left(88-44\sqrt{3}\right)}{25-3}}\)=\(\left(1+\sqrt{3}\right)\sqrt{\frac{22\left(4-2\sqrt{3}\right)}{22}}\)=\(\left(1+\sqrt{3}\right)\sqrt{\left(\sqrt{3}-1\right)^2}=\left(1+\sqrt{3}\right)\left(\sqrt{3}-1\right)\)=3-1 = 2
e) = \(\frac{\left(\sqrt{x}-1\right)\left(\sqrt{x}-3\right)}{\left(\sqrt{x}-3\right)\left(\sqrt{x}+3\right)}+\frac{7\sqrt{x}-3}{x-9}+\frac{\sqrt{x}\left(3-\sqrt{x}\right)}{3-\sqrt{x}}\)= \(\frac{x-4\sqrt{x}+3}{x-9}+\frac{7\sqrt{x}-3}{x-9}+\sqrt{x}\)= \(\frac{x+3\sqrt{x}}{x-9}+\sqrt{x}=\frac{\sqrt{x}\left(\sqrt{x}+3\right)}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}+\sqrt{x}\)= \(\frac{\sqrt{x}}{\sqrt{x}-3}+\sqrt{x}=\frac{x-2\sqrt{x}}{\sqrt{x}-3}\)
Tìm x ϵ Z ,để A ϵ Z
a)A=\(\frac{-\sqrt{x}+6}{\sqrt{x}-2}\)
b)A=\(\frac{4\sqrt{x}+1}{2\sqrt{x}-1}\)
c)A=\(\frac{\sqrt{x}-2}{3\sqrt{x}-4}\)
d)A=\(\frac{3\sqrt{x}}{5\sqrt{x}-1}\)
a) \(A=\frac{-\sqrt{x}+2+4}{\sqrt{x}-2}=-1+\frac{4}{\sqrt{x}-2}\)
Để \(A\in Z\Leftrightarrow\sqrt{x}-2\in\left\{-4;-2;-1;1;2;4\right\}\)
\(\Leftrightarrow\sqrt{x}\in\left\{-2;0;1;3;4;6\right\}\)
Mà \(x\in Z;\sqrt{x}\ge0\Rightarrow x\in\left\{0;1;9;16;36\right\}\)
b)\(A=\frac{4\sqrt{x}-2+3}{2\sqrt{x}-1}=2+\frac{3}{2\sqrt{x}-1}\)
Để \(A\in Z\Leftrightarrow2\sqrt{x}-1\in\left\{-3;-1;1;3\right\}\)
\(\Leftrightarrow2\sqrt{x}\in\left\{-2;0;2;4\right\}\)
\(\Leftrightarrow\sqrt{x}\in\left\{-1;0;1;2\right\}\Leftrightarrow x\in\left\{0;1;4\right\}\)
a) A= \(\frac{-\sqrt{x}+6}{\sqrt{x}-2}=\frac{-\sqrt{x}+2+4}{\sqrt{x}-2}=\frac{-\left(\sqrt{x}-2\right)+4}{\sqrt{x}-2}=\frac{4}{\sqrt{x}-2}-1\)
⇒ \(\sqrt{x}-2\inƯ\left(4\right)\) ⇒ x = 36
Đề hay nhỉ:) bấm vào câu hỏi thì ko thấy đề hai câu đầu, thoát ra ngoài trang hỏi đáp thì lại thấy;) công nhận người hỏi đăng hay:)
a) \(A=\frac{-\sqrt{x}+6}{\sqrt{x}-2}=\frac{-\sqrt{x}+2+4}{\sqrt{x}-2}=-1+\frac{4}{\sqrt{x}-2}\)
Do x thuộc Z nên để A nguyên \(\sqrt{x}-2\inƯ\left(4\right)=\left\{-2;-1;1;2;4\right\}\) (ko xét -4 vì \(\sqrt{x}-2\ge-2>-4\))
Rồi làm tiếp.
Sai thì em chịu
Cho A = \(\left(\dfrac{x-5\sqrt{x}}{x-25}-1\right):\left(\dfrac{25-x}{x+2\sqrt{x}-15}-\dfrac{\sqrt{x}+3}{\sqrt{x}+5}+\dfrac{\sqrt{x}-5}{\sqrt{x}-3}\right)\)
a ) Rút gọn A
b) Tìm x ϵ Z để A ϵ Z
a) Ta có: \(A=\left(\dfrac{x-5\sqrt{x}}{x-25}-1\right):\left(\dfrac{25-x}{x+2\sqrt{x}-15}-\dfrac{\sqrt{x}+3}{\sqrt{x}+5}+\dfrac{\sqrt{x}-5}{\sqrt{x}-3}\right)\)
\(=\left(\dfrac{\sqrt{x}\left(\sqrt{x}-5\right)}{\left(\sqrt{x}-5\right)\left(\sqrt{x}+5\right)}-1\right):\left(\dfrac{25-x}{\left(\sqrt{x}+5\right)\left(\sqrt{x}-3\right)}-\dfrac{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}{\left(\sqrt{x}+5\right)\left(\sqrt{x}-3\right)}+\dfrac{\left(\sqrt{x}-5\right)\left(\sqrt{x}+5\right)}{\left(\sqrt{x}+5\right)\left(\sqrt{x}-3\right)}\right)\)
\(=\left(\dfrac{\sqrt{x}}{\sqrt{x}+5}-1\right):\left(\dfrac{25-x-\left(x-9\right)+x-25}{\left(\sqrt{x}+5\right)\left(\sqrt{x}-3\right)}\right)\)
\(=\left(\dfrac{\sqrt{x}}{\sqrt{x}+5}-\dfrac{\sqrt{x}+5}{\sqrt{x}+5}\right):\left(\dfrac{25-x-x+9+x-25}{\left(\sqrt{x}+5\right)\left(\sqrt{x}-3\right)}\right)\)
\(=\dfrac{\sqrt{x}-\sqrt{x}-5}{\sqrt{x}+5}:\dfrac{x+9}{\left(\sqrt{x}+5\right)\left(\sqrt{x}-3\right)}\)
\(=\dfrac{-5}{\sqrt{x}+5}\cdot\dfrac{\left(\sqrt{x}+5\right)\left(\sqrt{x}-3\right)}{x+9}\)
\(=\dfrac{-5\left(\sqrt{x}-3\right)}{x+9}\)