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Ta có: \(\frac{\left(2007-x\right)^2+\left(2007-x\right)\left(x-2008\right)+\left(x-2008\right)^2}{\left(2007-x\right)^2-\left(2007-x\right)\left(2008-x\right)+\left(x-2008\right)^2}\)
\(=\frac{\left(2007-x\right)^2+\left(2007-x\right)\left(x-2008\right)+\left(x-2008\right)^2}{\left(2007-x\right)^2+\left(2007-x\right)\left(x-2008\right)+\left(x-2008\right)^2}\)
\(=1\)
Ta có : \(\frac{x^2-2008}{2007}+\frac{x^2-2007}{2006}+\frac{x^2-2006}{2005}=\frac{x^2-2005}{2004}+\frac{x^2-2004}{2003}+\frac{x^2-2003}{2002}\)
=> \(\frac{x^2-2008}{2007}+1+\frac{x^2-2007}{2006}+1+\frac{x^2-2006}{2005}+1=\frac{x^2-2005}{2004}+1+\frac{x^2-2004}{2003}+1+\frac{x^2-2003}{2002}+1\)
=> \(\frac{x^2-2008}{2007}+\frac{2007}{2007}+\frac{x^2-2007}{2006}+\frac{2006}{2006}+\frac{x^2-2006}{2005}+\frac{2005}{2005}=\frac{x^2-2005}{2004}+\frac{2004}{2004}+\frac{x^2-2004}{2003}+\frac{2003}{2003}+\frac{x^2-2003}{2002}+\frac{2002}{2002}\)
=> \(\frac{x^2-1}{2007}+\frac{x^2-1}{2006}+\frac{x^2-1}{2005}=\frac{x^2-1}{2004}+\frac{x^2-1}{2003}+\frac{x^2-1}{2002}\)
=> \(\frac{x^2-1}{2007}+\frac{x^2-1}{2006}+\frac{x^2-1}{2005}-\frac{x^2-1}{2004}-\frac{x^2-1}{2003}-\frac{x^2-1}{2002}=0\)
=> \(\left(x^2-1\right)\left(\frac{1}{2007}+\frac{1}{2006}+\frac{1}{2005}-\frac{1}{2004}-\frac{1}{2003}-\frac{1}{2002}\right)=0\)
=> \(x^2-1=0\)
=> \(x^2=1\)
=> \(x=\pm1\)
Vậy phương trình có 2 nghiệm là x = 1, x = -1 .
\(\frac{x+2}{2008}+\frac{x+3}{2007}+\frac{x+4}{2006}+\frac{x+2028}{6}=0\)
\(\Rightarrow\left(\frac{x+2}{2008}+1\right)+\left(\frac{x+3}{2007}+1\right)+\left(\frac{x+4}{2006}+1\right)+\left(\frac{x+2028}{6}-3\right)=0\)
\(\Rightarrow\frac{x+2010}{2008}+\frac{x+2010}{2007}+\frac{x+2010}{2006}+\frac{x+2010}{6}=0\)
\(\Rightarrow\left(x+2010\right)\left(\frac{1}{2008}+\frac{1}{2007}+\frac{1}{2006}+\frac{1}{6}\right)=0\)
Mà \(\frac{1}{2008}+\frac{1}{2007}+\frac{1}{2006}+\frac{1}{6}\ne0\)
\(\Rightarrow x+2010=0\)
\(\Rightarrow x=-2010\)
Vậy x = -2010
\(\dfrac{x+2}{2008}+\dfrac{x+3}{2007}+\dfrac{x+4}{2006}+\dfrac{x+2028}{6}=0\)
\(\Leftrightarrow\dfrac{x+2}{2008}+1+\dfrac{x+3}{2007}+1+\dfrac{x+4}{2006}+1+\dfrac{x+2028}{6}-3=0\)
\(\Leftrightarrow\dfrac{x+2010}{2008}+\dfrac{x+2010}{2007}+\dfrac{x+2010}{2006}+\dfrac{x+2010}{6}=0\)
\(\Leftrightarrow\left(x+2010\right)\left(\dfrac{1}{2008}+\dfrac{1}{2007}+\dfrac{1}{2006}+\dfrac{1}{6}\right)=0\)
\(\Leftrightarrow x+2010=0\). Do \(\dfrac{1}{2008}+\dfrac{1}{2007}+\dfrac{1}{2006}+\dfrac{1}{6}\ne0\)
\(\Leftrightarrow x=-2010\)
\(\frac{x+2}{2008}\)+ 1 + \(\frac{x+3}{2007}\)+1 +\(\frac{x+4}{2006}\)+1 +\(\frac{x+2028}{6}\)-3=0
\(\Leftrightarrow\)\(\frac{x+2+2008}{2008}\)+ \(\frac{x+3+2007}{2007}\) + \(\frac{x+4+2006}{2006}\)+ \(\frac{x+2028-18}{6}\)= 0
\(\Leftrightarrow\) \(\frac{x+2010}{2008}\)+ \(\frac{x+2010}{2007}\)+ \(\frac{x+2010}{2006}\)+ \(\frac{x+2010}{6}\)=0
\(\Leftrightarrow\)(x +2010).\(\left(\frac{1}{2008}+\frac{1}{2007}+\frac{1}{2006}+\frac{1}{6}\right)\)=0
\(\Leftrightarrow\)x + 2010 = 0 \(\left(vì\frac{1}{2008}+\frac{1}{2007}+\frac{1}{2006}+\frac{1}{6}>0\right)\)
\(\Leftrightarrow\) x = -2010
Vậy S = \(\left\{-2010\right\}\)
⇔ \(\dfrac{x+2}{2008}\) +1 +\(\dfrac{x+3}{2007}\) +1+\(\dfrac{x+4}{2006}\)+1 +\(\dfrac{2028}{6}\)-3 =0
⇔\(\dfrac{x+2}{2008}+\dfrac{2008}{2008}+\dfrac{x+3}{2007}+\dfrac{2007}{2007}+\dfrac{x+4}{2006}+\dfrac{2006}{2006}+\dfrac{x+2028}{6}-\dfrac{18}{6}=0\)
⇔\(\dfrac{x+2010}{2008}+\dfrac{x+2010}{2007}+\dfrac{x+2010}{2006}+\dfrac{x+2010}{6}=0\)
⇔(x+2010)\(\left(\dfrac{1}{2008}+\dfrac{1}{2007}+\dfrac{1}{2006}+\dfrac{1}{6}\right)=0\)
Mà \(\left(\dfrac{1}{2008}+\dfrac{1}{2007}+\dfrac{1}{2006}+\dfrac{1}{6}\right)\)≠0
⇒x+2010=0
⇔x=-2010
Vậy phương trình có nghiệm x=-2010
\(\frac{\left(2007-x\right)^2+\left(2007-x\right)\left(x-2008\right)+\left(x-2008\right)^2}{\left(2007-x\right)^2-\left(2007-x\right)\left(x-2008\right)+\left(x-2008\right)^2}=\frac{19}{49}\)
điểu kiện xác định x khác 2007 and x khác 2008
Đặt a=x-2008 ( a khác 0 ,) ta có hệ thức
\(\frac{\left(a+1\right)^2-\left(a+1\right)a+a^2}{\left(a+1\right)^2+\left(a+1\right)a+a^2}=\frac{19}{49}\)
=>\(\frac{a^2+a+1}{3a^2+3a+1}=\frac{19}{49}\)
=>\(49a^2+49a+49=57a^2+57a+19\)
=>\(8a^2+8a-30=0\)
=>\(\left(2a-1\right)^2-4^2=0=>\left(2a-3\right)\left(2a+5\right)=0\)
=>\(\orbr{\begin{cases}a=\frac{3}{2}\\a=-\frac{5}{2}\end{cases}}\)(Thỏa mãn điều kiện)
Tự thay a xong suy ra x nhá
Mệt lắm r
\(\frac{x+2}{2008}+\frac{x+3}{2007}+\frac{x+4}{2006}+\frac{x+2028}{6}=0\)
\(\Leftrightarrow\left(\frac{x+2}{2008}+1\right)+\left(\frac{x+3}{2007}+1\right)+\left(\frac{x+4}{2006}+1\right)+\frac{x+2010}{6}=0\)
\(\Leftrightarrow\frac{x+2010}{2008}+\frac{x+2010}{2007}+\frac{x+2010}{2006}+\frac{x+2010}{6}=0\)
\(\Leftrightarrow\left(x+2010\right)\left(\frac{1}{2008}+\frac{1}{2007}+\frac{1}{2006}+\frac{1}{6}\right)=0\)
\(\Rightarrow x+2010=0\Rightarrow x=-2010\)
\(A=x^6-2007x^5+2007x^4-2007x^3+2007x^2-2007x+2007\)
\(=x^6-2006x^5-x^5+2006x^4+x^4-2006x^3-x^3+2006x^2+x^2-2006x-x+2006+1\)
\(=x^5\left(x-2006\right)-x^4\left(x-2006\right)+x^3\left(x-2006\right)-x^2\left(x-2006\right)+x\left(x-2006\right)-\left(x-2006\right)+1\)
\(=\left(x^5-x^4+x^3-x^2+x-1\right)\left(x-2006\right)+1\)
Thay x = 2006
\(\Leftrightarrow A=1\)
Vậy A = 1 tại x = 2006
\(A=x^6-2007.x^5+2007.x^4-2007.x^3+2007.x^2-2007.x+2007\)
\(=x^6-\left(x+1\right).x^5+\left(x+1\right).x^4-...+x+1\)
\(=x^6-x^6-x^5+x^5+x^4-x^4-...-x+1\)
\(=1\)