1. Cho A=\(\frac{3}{2+\sqrt{2x-x^2}+3}\)

a. Tìm x để A có nghĩa

b. Tìm Min(A), Max(A)

2/ Tìm Min, Max của:    \(A=\frac{1}{2+\sqrt{x-x^2}}\)

3/ Tìm Min(B) biết: \(B=\sqrt{x+2\sqrt{x-1}}+\sqrt{x-2\sqrt{x-1}}\)

4/ Tìm Min, Max của:\(C=\frac{4x+3}{x^2+1}\)

5/ Tìm Max của: \(A=\sqrt{x-1}+\sqrt{y-2}\)biết \(x+y=4\)

6/ Tìm Max(B) biết: \(B=\frac{y\sqrt{x-1}+x\sqrt{y-2}}{xy}\)

7/ Tìm Max(C) biết: \(C=x+\sqrt{2-x}\)

giải phương trình

a) \(\left(x+\frac{5-x}{\sqrt{x}+1}\right)^2+\frac{16\sqrt{x}\left(5-x\right)}{\sqrt{x}+1}-16\)\(=0\)

b) \(\sqrt{2x-\frac{3}{x}}+\sqrt{\frac{6}{x}-2x}=1+\frac{3}{2x}\)

c) \(\sqrt{2x+1}+\frac{2x-1}{x+3}-\left(2x-1\right)\sqrt{x^2+4}-\sqrt{2}=0\)

d) \(\sqrt{x+2+3\sqrt{2x-5}}+\sqrt{x-2-\sqrt{2x-5}}=2\sqrt{2}\)

Tìm GTNN, GTLN của các:

a. \(A=\dfrac{1}{5+2\sqrt{6-x^2}}\)

b. \(B=\sqrt{-x^2+2x+4}\)

c. \(C=\left|x-\sqrt{2}\right|+\left|y-1\right|\) trong đó \(\left|x\right|+\left|y\right|=5\)

d. \(D=\sqrt{1-x}+\sqrt{1+x}\)

e. \(E=\sqrt{x-2}+\sqrt{6-x}\)

f. \(F=2x+\sqrt{5-x^2}\)

g. \(G=x\left(99+\sqrt{101-x^2}\right)\)

h. \(H=x^2+y^2\) biết rằng \(x^2\left(x^2+2y^2-3\right)+\left(y^2-2\right)^2=1\)

i. \(I=x+y+z+xy+yz+zx\) biết rằng \(x^2+4y^2=1\)

k. \(K=x^3+y^3\) biết rằng \(x\ge0,y\ge0,x^2+y^2=1\)

l. \(L=x\sqrt{x}+y\sqrt{y}\) biết \(\sqrt{x}+\sqrt{y}=1\)

m. \(M=x\left(x^2-6\right)\) biết \(0\le x\le3\)

1.Tìm x để biểu thức sau có nghĩa

a) \(A=\sqrt{\frac{2x+3}{x-3}}\)

b) \(B=\frac{\sqrt{2x+3}}{\sqrt{x-3}}\)

c)\(C=\sqrt{-\frac{5}{x+2}}\)

d) \(D=\sqrt{-x}+\frac{1}{x+3}\)

2. Rút gọn biểu thức

a) \(A=\sqrt{\frac{a+\sqrt{a^2-1}}{2}}+\sqrt{\frac{a-\sqrt{a^2-1}}{2}};\left(a>1\right)\)

b) \(B=\sqrt{\frac{a+\sqrt{a^2-b}}{2}}-\sqrt{\frac{a-\sqrt{a^2-b}}{2}}\)( \(a\) > \(\sqrt{b}\); b > \(0\))

c)\(C=\left(1+\frac{a+\sqrt{a}}{\sqrt{a+1}}\right)\left(1-\frac{a-\sqrt{a}}{\sqrt{a}-1}\right)\)( a > 0 ; a\(\ne\)1 )

d) \(D=\frac{x^2+\sqrt{x}}{x-\sqrt{x}+1}-\frac{2x+\sqrt{x}}{\sqrt{x}};\left(x>0\right)\)

Bài 2
a) A= \(\sqrt{\left(1-\sqrt{2}\right)^2}+\sqrt{\left(-2\right)^6}-\sqrt{\left(1+\sqrt{2}\right)^2}\)

b) B= \(\sqrt{7+2\sqrt{6}}+\sqrt{7-2\sqrt{6}}\)
c) C= \(\sqrt{7-4\sqrt{3}}\)

d) D= \(2\sqrt{7+4\sqrt{3}}-\sqrt{13-4\sqrt{3}}\)

e) E= \(\frac{1}{1+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}+...+\frac{1}{\sqrt{79}+\sqrt{81}}\)
Bài 4:
a) \(\sqrt{x-1}=2\)

b) \(\sqrt{x^2-3x+2}=\sqrt{2}\)

c) \(\sqrt{4x+1}=x+1\)

d) \(\sqrt{x+2\sqrt{x-1}}-\sqrt{x-2\sqrt{x-1}}=2\)

e) \(\sqrt{x^2-4x+5}+\sqrt{x^2-4x+8}+\sqrt{x^2-4x+9}=3+\sqrt{5}\)

f)

Rút gọn

a) \(\sqrt{75}+\sqrt{48}-\)\(\sqrt{192}\)

b)\(3\sqrt{2x}-5\sqrt{2x}-5\sqrt{2x}=9-6\sqrt{2x}\left(x>0\right)\)

c)\(3\sqrt{2x}-4\sqrt{8x}-5\sqrt{50x}\left(x>0\right)\)

d)\(\frac{1}{x^2-y^2}.\sqrt{\frac{2\left(x+y\right)^2}{3}}\left(x\ge0;y\ge0;x\ne y\right)\)

e)\(\left(3\sqrt{2}+\sqrt{3}\right).\sqrt{2}-\sqrt{54}\)

f)\(2\sqrt{21}-\left(\sqrt{28}+\sqrt{12}-\sqrt{7}\right).\sqrt{7}\)
 

a. \(\sqrt{1-2x}=x+7\)

b.\(\sqrt{1+x^2}=2x-1\)

c.\(\sqrt{1-x}-\sqrt{2+x}=1\)

d. \(\sqrt{1-x}+\sqrt{4+x}=3\)

e. \(\left|2x+5\right|\)= x-1

f. \(\left|x\right|+\left|2x-1\right|=x+2\)

g.\(\left|x-9\right|^{10}+\left|x-10\right|^{10}=1\)

h. \(\left|x-2014\right|^{2015}+\left|x-2015\right|^{2014}=1\)

i. \(\sqrt{x+3+4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=5\)

k. \(\sqrt{x+2\sqrt{x-1}}+\sqrt{x-2\sqrt{x-1}}=x-1\)

l. \(\sqrt{x+2-4\sqrt{x-2}}+\sqrt{x+7-6\sqrt{x-2}}=1\)

m. \(\sqrt{x+8+2\sqrt{x+7}}+\sqrt{x-1-2\sqrt{x+7}}=4\)

m. \(x+\sqrt{x+\frac{1}{4}+\sqrt{x+\frac{1}{4}}}=\frac{9}{4}\)

n.\(\left(x+5\right)^4+\left(x+7\right)^4=2\)

Giải pt

Rút gọn

a)\(\sqrt{75}+\sqrt{75}-\)\(\sqrt{192}\)

b)3\(\sqrt{2x}-5\sqrt{2x}-5\sqrt{2x}+9-6\sqrt{2x}\left(x>0\right)\)

c)3\(\sqrt{2x}-4\sqrt{8x}-5\sqrt{50x}\left(x>0\right)\)

d)\(\frac{1}{x^2-y^2}.\sqrt{\frac{2\left(x+y\right)^2}{3}}\left(x\ge0;y\ge0;x\ne y\right)\)

e)\(\left(3\sqrt{2}+\sqrt{3}\right).\sqrt{2}\sqrt{54}\)

f)\(2\sqrt{21}-\left(\sqrt{28}+\sqrt{12}-\sqrt{7}\right).\sqrt{7}\)

Gpt: a) \(\sqrt[4]{3\left(x+5\right)}-\sqrt[4]{11-x}=\sqrt[4]{13+x}-\sqrt[4]{3\left(3-x\right)}\)

b) \(\frac{1+2\sqrt{x}-x\sqrt{x}}{3-x-\sqrt{2-x}}=2\left(\frac{1+x\sqrt{x}}{1+x}\right)\) c) \(\sqrt{x+1}+\frac{4\left(\sqrt{x+1}+\sqrt{x-2}\right)}{3\left(\sqrt{x-2}+1\right)^2}=3\)

d) \(\sqrt{\frac{x-2}{x+1}}+\frac{x+2}{\left(\sqrt{x+2}+\sqrt{x-2}\right)^2}=1\) e) \(2x+1+x\sqrt{x^2+2}+\left(x+1\right)\sqrt{x^2+2x+2}=0\)

f) \(\sqrt{2x+3}\cdot\sqrt[3]{x+5}=x^2+x-6\)