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b)Ta có: \(a^{2000}+b^{2000}=a^{2001}+b^{2001}\)
\(\Rightarrow a^{2001}+b^{2001}\)\(-a^{2000}-b^{2000}=0\)
\(\Rightarrow a^{2000}\left(a-1\right)+b^{2000}\left(b-1\right)=0\)(1)
và \(a^{2001}+b^{2001}=a^{2002}+b^{2002}\)
\(\Rightarrow a^{2002}+b^{2002}\)\(-a^{2001}-b^{2001}=0\)
\(\Rightarrow a^{2001}\left(a-1\right)+b^{2001}\left(b-1\right)=0\)(2)
Lấy (2) - (1), ta được: \(a^{2000}\left(a-1\right)^2+b^{2000}\left(b-1\right)^2=0\)(3)
Mà \(a^{2000}\left(a-1\right)^2\ge0\forall a\)và \(b^{2000}\left(b-1\right)^2\ge0\forall b\)
nên (3) xảy ra\(\Leftrightarrow\hept{\begin{cases}a^{2000}\left(a-1\right)^2=0\\b^{2000}\left(b-1\right)^2=0\end{cases}}\Leftrightarrow\hept{\begin{cases}a=1hoaca=0\\b=1hoacb=0\end{cases}}\)
Mà a,b dương nên a = 1 và b = 1
a) Áp dụng BĐT Svac - xơ:
\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge\frac{\left(1+1+1\right)^2}{a+b+c}=9\)
(Dấu "="\(\Leftrightarrow a=b=c=\frac{1}{3}\))
a: \(\Leftrightarrow2\sqrt{3x}+12-4x+5\sqrt{3}=0\)
\(\Leftrightarrow-4x+2\sqrt{3}\cdot\sqrt{x}+12+5\sqrt{3}=0\)
Đặt \(\sqrt{x}=a\left(a>=0\right)\)
Phương trình trở thành \(-4a^2+2\sqrt{3}a+12+5\sqrt{3}=0\)
\(\Delta=\left(2\sqrt{3}\right)^2-4\cdot\left(-4\right)\cdot\left(12+5\sqrt{3}\right)\)
\(=12+16\left(12+5\sqrt{3}\right)\)
\(=12+192+80\sqrt{3}=204+80\sqrt{3}\)
Vì Δ>0 nên phương trình có hai nghiệm phân biệt là:
\(\left\{{}\begin{matrix}a_1=\dfrac{-2\sqrt{3}-\sqrt{204+80\sqrt{3}}}{-8}=\dfrac{2\sqrt{3}+\sqrt{204+80\sqrt{3}}}{8}\left(nhận\right)\\a_2=\dfrac{-2\sqrt{3}+\sqrt{204+80\sqrt{3}}}{-8}\left(loại\right)\end{matrix}\right.\)
\(\Leftrightarrow a=\dfrac{2\sqrt{3}+2\sqrt{26+20\sqrt{3}}}{8}=\dfrac{\sqrt{3}+\sqrt{26+20\sqrt{3}}}{4}\)
\(\Leftrightarrow x=a^2\simeq5,66\)
c: \(\Leftrightarrow x\sqrt{2}+5\sqrt{2}-4x-5-4\sqrt{2}=0\)
\(\Leftrightarrow x\left(\sqrt{2}-4\right)+\sqrt{2}-5=0\)
\(\Leftrightarrow x=\dfrac{5-\sqrt{2}}{\sqrt{2}-4}=\dfrac{-18-\sqrt{2}}{14}\)
d: \(\Leftrightarrow\dfrac{7x+1-4x-4002}{2001}=\dfrac{3x+2}{2003}-1\)
\(\Leftrightarrow3x-4001=0\)
hay x=4001/3
\(\Rightarrow\frac{x+4}{2000}+1+\frac{x+3}{2001}+1=\frac{x+2}{2002}+1+\frac{x+1}{2013}+1\)
\(\Leftrightarrow\frac{x+2004}{2000}+\frac{x+2004}{2001}=\frac{x+2004}{2002}+\frac{x+2004}{2003}\)
\(\Leftrightarrow\frac{x+2004}{2000}+\frac{x+2004}{2001}-\frac{x+2004}{2002}-\frac{x+2004}{2003}=0\)
\(\Leftrightarrow\left(x+2004\right)\left(\frac{1}{2000}+\frac{1}{2001}-\frac{1}{2002}-\frac{1}{2003}\right)=0\)
=> x +2004 =0 ( 1/2000 + 1/2001 - 1/2002 - 1/2003 khác 0 )
=> x = -2004
Vậy x = -2004 là nghiệm của pt
\(\frac{x+4}{2000}+\frac{x+3}{2001}=\frac{x+2}{2002}+\frac{x+1}{2003}\)
\(\Leftrightarrow\frac{x+4}{2000}+\frac{2000}{2000}+\frac{x+3}{2001}+\frac{2001}{2001}=\frac{x+2}{2002}+\frac{2002}{2002}+\frac{x+1}{2003}+\frac{2003}{2003}\)
\(\Leftrightarrow\frac{x+2004}{2000}+\frac{x+2004}{2001}-\frac{x+2004}{2002}-\frac{x+2004}{2003}=0\)
\(\Leftrightarrow\left(x+2004\right).\left(\frac{1}{2000}+\frac{1}{2001}-\frac{1}{2002}-\frac{1}{2003}\right)=0\)
mà \(\frac{1}{2000}+\frac{1}{2001}-\frac{1}{2002}-\frac{1}{2003}\ne0\)nên
x+2004=0
<=>x=-2004
Theo mình thì câu 2 là :
a/ b+c + b/c+a + c/a+b =1
suy ra (a+b+c) * (a/ b+c + b/c+a + c/a+b ) = a+b+c
suy ra a*(a+b+c)/(b+c) + b*(a+b+c)/(c+a) + c*(a+b+c)/(a+b) = a+b+c
suy ra a^2+a*(b+c)/b+c +b^2 +b*(c+a)/ c+a +c^2+c*(a+b)/a+b =a=b+c
suy ra a^2/(b+c) +a +b^2/(c+a) +b +c^2/(a+b) +c =a+b+c
suy ra a^2/(b+c) +b^2/(c+a) +c^2/(a+b) =a+b+c -a-b-c
suy ra a^2/(b+c) +b^2/(c+a) +c^2/(a+b) = 0
\(a.\frac{x-23}{24}+\frac{x-23}{25}=\frac{x-23}{26}+\frac{x-23}{27}\\\Leftrightarrow \left(x-23\right)\left(\frac{1}{24}+\frac{1}{25}-\frac{1}{26}-\frac{1}{27}\right)=0\\\Leftrightarrow x-23=0\left(vi\frac{1}{24}+\frac{1}{25}-\frac{1}{26}-\frac{1}{27}\ne0\right)\\ \Leftrightarrow x=23\)
Này tớ làm tắt có gì cậu không hiểu nói tớ nhé
\(b.\left(\frac{x+2}{98}+1\right)+\left(\frac{x+3}{97}+1\right)=\left(\frac{x+4}{96}+1\right)+\left(\frac{x+5}{95}+1\right)\\ \Leftrightarrow\frac{x+2}{98}+1+\frac{x+3}{97}+1-\left(\frac{x+4}{96}+1+\frac{x+5}{95}+1\right)=0\\\Leftrightarrow \frac{x+100}{98}+\frac{x+100}{97}-\frac{x+100}{96}-\frac{x+100}{95}=0\\\Leftrightarrow \left(x+100\right)\left(\frac{1}{98}+\frac{1}{97}-\frac{1}{96}-\frac{1}{95}\right)=0\\ \Leftrightarrow x+100=0\left(Vi\frac{1}{98}+\frac{1}{97}-\frac{1}{96}-\frac{1}{95}\ne0\right)\\\Leftrightarrow x=-100\)
\(C=\frac{1999.4001+2000}{2000.4001-2001}\)
\(\Leftrightarrow C=\frac{\left(2000-1\right).4001+2000}{2000.4001-2001}\)
\(\Leftrightarrow C=\frac{2000.4001-4001+2000}{2000.4001-2001}\)
\(\Leftrightarrow C=\frac{2000.4001-2001}{2000.4001-2001}=1\)
\(D=\frac{1501.1503-1500.1498}{6002}\)
\(\Leftrightarrow D=\frac{\left(1502-1\right)\left(1502+1\right)-\left(1499+1\right)\left(1498-1\right)}{6002}\)
\(\Leftrightarrow D=\frac{1502^2-1-1499^2+1}{6002}\)
\(\Leftrightarrow D=\frac{\left(1502-1499\right)\left(1502+1499\right)}{6002}\)
\(\Leftrightarrow D=\frac{3.3001}{6002}=\frac{3.3001}{2.3001}=\frac{3}{2}\)
So sánh 1 và 3/2, Ta thấy C<D
\(C=\frac{\left(3000-1001\right)\left(3000+1001\right)+2000}{2000\left(4001-1\right)+1}\)
\(=\frac{3000^2-1001^2+2000}{2000\cdot4000+1}=\frac{9000000-1002001+2000}{8000000+1}\)
\(=\frac{7999999}{8000001}=0,99999975\)
còn câu D mk làm ở câu khác rồi nên mk ghi luôn kết quả nha
D = 1,5
=> C < D