\(\sqrt{x^2-2002x+2001}+\sqrt{x^2-2003x+2002}=\sqrt{x^2-2004x+2003}\)
Cho \(\left(x+\sqrt{x^2+2003}\right)\cdot\left(y+\sqrt{y^2+2003}\right)=2003\)
Tính x+y?
1. Tính giá trị biểu thức: \(A=\sqrt{a^2+4ab^2+4b}-\sqrt{4a^2-12ab^2+9b^4}\) với \(a=\sqrt{2}\) ; \(b=1\)
2. Đặt \(M=\sqrt{57+40\sqrt{2}}\) ; \(N=\sqrt{57-40\sqrt{2}}\). Tính giá trị của các biểu thức sau:
a) M-N
b) \(M^3-N^3\)
3. Chứng minh: \(\left(\frac{x\sqrt{x}+3\sqrt{3}}{x-\sqrt{3x}+3}-2\sqrt{x}\right)\left(\frac{\sqrt{x}+\sqrt{3}}{3-x}\right)=1\) (với \(x\ge0\) và \(x\ne3\))
4. Chứng minh: \(\frac{\left(\sqrt{a}-\sqrt{b}\right)^2+4\sqrt{ab}}{\sqrt{a}+\sqrt{b}}.\frac{a\sqrt{b}-b\sqrt{a}}{\sqrt{ab}}=a-b\) (a > 0 ; b > 0)
5. Chứng minh: \(\sqrt{9+4\sqrt{2}}=2\sqrt{2}+1\) ; \(\sqrt{13+30\sqrt{2+\sqrt{9+4\sqrt{2}}}}=5+3\sqrt{2}\) ; \(3-2\sqrt{2}=\left(1-\sqrt{2}\right)^2\)
6. Chứng minh: \(\left(\frac{1}{2\sqrt{2}-\sqrt{7}}-\left(3\sqrt{2}+\sqrt{17}\right)\right)^2=\left(\frac{1}{2\sqrt{2}-\sqrt{17}}-\left(2\sqrt{2}-\sqrt{17}\right)\right)^2\)
7. Chứng minh đẳng thức: \(\left(\frac{3\sqrt{2}-\sqrt{6}}{\sqrt{27}-3}-\frac{\sqrt{150}}{3}\right).\frac{1}{\sqrt{6}}=-\frac{4}{3}\)
8.Chứng minh: \(\frac{2002}{\sqrt{2003}}+\frac{2003}{\sqrt{2002}}>\sqrt{2002}+\sqrt{2003}\)
9. Chứng minh rằng: \(\sqrt{2000}-2\sqrt{2001}+\sqrt{2002}< 0\)
10. \(\frac{1}{2}+\frac{1}{3\sqrt{2}}+...+\frac{1}{\left(n+1\right)\sqrt{n}}< 2\) ; \(\frac{7}{5}< \frac{2+\sqrt{3}}{\sqrt{2}+\sqrt{2+\sqrt{3}}}+\frac{2-\sqrt{3}}{\sqrt{2}-\sqrt{2-\sqrt{3}}}< \frac{29}{30}\)
Giai he pt: \(\left\{{}\begin{matrix}\left(x-y\right)^2+4=3y-5x+2\sqrt{\left(x+1\right)\left(y-1\right)}\\\frac{3xy-5y-6x+11}{\sqrt{x^3+1}}=5\end{matrix}\right.\)
giai pt\(\dfrac{x^2}{\left(\sqrt{x+1}+1\right)^2}=x-4\)
giai pt :
\(x^2+4x+7=\left(x+4\right)\sqrt{x^2+7}\)
Giai PT \(\sqrt{x-\frac{1}{x}}-\sqrt{1-\frac{1}{x}}=\frac{x-1}{x}\)
Giai pt:
a)\(\sqrt{4x}=\sqrt{5}\)
b)\(\sqrt{4\left(1-x\right)^2}-6=0\)
c)\(\sqrt{x^2-4x+4}=3\)
Giải:
1)a) \(17\sqrt{3x-1}=3x\)
b) \(\sqrt{2+\sqrt{3x-5}}=\sqrt{x+1}\)
c)\(\sqrt{\dfrac{5x+7}{x+3}}=4\)
2)Giai pt :
a) x+y+12=\(4\sqrt{x}+6\sqrt{y}-1\)
b) \(\sqrt{x-a}+\sqrt{y-b}+\sqrt{z-c}=\dfrac{1}{2}\left(x+y+z\right)\)
c)\(\sqrt{x+\sqrt{14x-4y}}+\sqrt{x-\sqrt{14x-4y}}=\sqrt{14}\)
d)x-\(4\sqrt{2x+2}-2\sqrt{2-x+9}=0\)
