D= \(\frac{x^2-2x+2000}{x^2}\)= 1-\(\frac{2x}{x^2}\)+\(\frac{2000}{x^2}\)

Đặt t= \(\frac{1}{2}\)=> D= 2000t²-2t+1 = (\(20\sqrt{5}t\))²-2.\(20\sqrt{5}t\).\(\frac{1}{20\sqrt{5}}\)+\(\left(\frac{1}{20\sqrt{5}}\right)^2\)\(-\left(\frac{1}{20\sqrt{5}}\right)^2\)+1

D= (\(20\sqrt{5}t\)-\(\frac{1}{20\sqrt{5}}\)) ²+\(\frac{1999}{2000}\)\(\ge\)\(\frac{1999}{2000}\)

Min D= \(\frac{1999}{2000}\)khi \(20\sqrt{5}t\)\(-\frac{1}{20\sqrt{5}}\)= 0 => t = \(\frac{1}{2000}\)=> \(\frac{1}{x}\)= \(\frac{1}{2000}\)=> x= 2000

Bai 1: Tinh

a) ( \(\frac{1}{3}\))^-1 - (\(\frac{-6}{7}\))⁰ + ( \(\frac{1}{2}\))^{-2 .} 2

b) \(\frac{2.5^{22}-9.5^{21}}{25^{10}}\): \(\frac{5.\left(3.7^{15}-19.7^{14}\right)}{7^{16}+3.7^{15}}\)

c) ( \(\frac{-4}{9}\))³. (\(\frac{3}{2}\))². (\(\frac{9}{6}\))³

B= \(\frac{x-4x+1}{x^2}\)= 1-\(\frac{4}{x}\)+\(\frac{1}{x^2}\)

Đặt t = \(\frac{1}{x}\)=> B= 1-4t+t²= t²-4t+1 = t²-2.t.2+4-3= (t-2)²-3 \(\ge\)-3 

Min của A= -3 khi t-2= 0 => t=2 => \(\frac{1}{x}\)= 2 => x = \(\frac{1}{2}\)

A =5+ 5²+5³+..................................................+5¹⁰⁰

B= 5+5³ +5⁵ +...................................................+5⁹⁹

C= 1+ \(\frac{1}{3}\) + \(\frac{1}{3^2}\) +....................................+ \(\frac{1}{3^{100}}\)

D= 1-\(\frac{1}{3}\)+ \(\frac{1}{3^2}\) - ..........................................- \(\frac{1}{3^{99}}\)+\(\frac{1}{3^{100}}\)

E=\(\frac{1}{2}\)+\(\frac{2}{2^2}\)+\(\frac{3}{2^3}\) +................................... +\(\frac{100}{2^{100}}\)

giải pt , \(\sqrt{x^4+4x^2}+\sqrt{x+x^2}=\sqrt{\left(x^2+\sqrt{x}\right)^2+9x^2}.\)

\(x=0\)

\(x^3=0\)

\(x^3=2.0.\sqrt{0}\)

\(x^3=2x\sqrt{x}\)

\(x^3=2x\sqrt{x}\)     

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

\(4\left(x^6-4x^4\sqrt{x}+4x^2x\right)=0\)

\(4x^6-16x^4\sqrt{x}+16x^2x=0\)

\(4x^6+16x^3=16x^4\sqrt{x}\)

\(16x^4+4x^5+4x^6+16x^3=16x^4+4x^5+16x^4\sqrt{x}\)

\(4x^3\left(x+1\right)\left(x^2+4\right)=4\left(4x^4+4x^4\sqrt{x}+x^4.x\right)\)

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

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

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

\(\left(\sqrt{x^4+4x^2}+\sqrt{x^2+x}\right)^2=\left(x^4+2x^2\sqrt{x}+x\right)+9x^2\)

\(\sqrt{x^4+4x^2}+\sqrt{x^2+x}=\sqrt{\left(x^2+\sqrt{x}\right)^2+9x^2}\)

vậy x=0 là nghiệm của pt  =))