Important Middle term

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1.  x^{2}-2000\frac{2000}{2021}x-1=0 

উত্তরঃ 

x^{2}-2000\frac{2000}{2001}x-1=0 
ধরি , 2001 = a , \Rightarrow 2000 = a -1

 
    x^{2}-2000\frac{2000}{2001}x-1=0\\\\\Rightarrow x^{2}-[(a-1)+\frac{a-1}{a}]x-1=0\\\\\Rightarrow x^{2}-[\frac{a(a-1)+(a-1)}{a}]x-1=0\\\\\Rightarrow x^{2}-\frac{a^{2}-1}{a}x-1=0\\\\\Rightarrow ax^{2}-(a^{2}-1)x-a=0\\\\\Rightarrow ax^{2}-a^{2}x+x-a=0\\\\\Rightarrow ax(x-a)+1(x-a)=0\\\\\Rightarrow (x-a)(ax+1)=0
     \Rightarrow (x-a)=0  অথবা (xa+1)=0        \Rightarrow x=a
  অথবা  x=-\frac{1}{a} 
    \therefore x= 2001  অথবা x= -\frac{1}{2001} 

2. x^{2}-65\frac{32}{33}x-2=0

উত্তরঃ 

      x^{2}-65\frac{32}{33}x-2=0 
ধরি , 33 = a

, 

\Rightarrow 32 = a -1, এবং 65=2a-1 
    x^{2}-65\frac{32}{33}x-2=0\\\\\Rightarrow x^{2}-[(2a-1)+\frac{a-1}{a}]x-2=0\\\\\Rightarrow x^{2}-[\frac{a(2a-1)+(a-1)}{a}]x-2=0\\\\\Rightarrow x^{2}-\frac{2a^{2}-1}{a}x-2=0\\\\\Rightarrow ax^{2}-(2a^{2}-1)x-2a=0\\\\\Rightarrow ax^{2}-2a^{2}x+x-2a=0\\\\\Rightarrow ax(x-2a)+1(x-2a)=0\\\\\Rightarrow (x-2a)(ax+1)=0
     \Rightarrow (x-2a)=0  অথবা (xa+1)=0

     

  \Rightarrow x=2a  অথবা  x=-\frac{1}{a} 
    \therefore x= 66  অথবা x= -\frac{1}{33} 

3. সমাধান করো :  x-\frac{1}{x}=38\frac{38}{39} 

\\x-\frac{1}{x}=38\frac{38}{39}\\\\\Rightarrow \frac{x^{2}-1}{x}=38\frac{38}{39}\\\\\Rightarrow x^{2}-1=38\frac{38}{39}x\\\\\Rightarrow x^{2}-38\frac{38}{39}x-1=0 
ধরি , 39 = a ,

 \Rightarrow 38 = a -1 
    x^{2}-38\frac{38}{39}x-1=0\\\\\Rightarrow x^{2}-[(a-1)+\frac{a-1}{a}]x-1=0\\\\\Rightarrow x^{2}-[\frac{a(a-1)+(a-1)}{a}]x-1=0\\\\\Rightarrow x^{2}-\frac{a^{2}-1}{a}x-1=0\\\\\Rightarrow ax^{2}-(a^{2}-1)x-a=0\\\\\Rightarrow ax^{2}-a^{2}x+x-a=0\\\\\Rightarrow ax(x-a)+1(x-a)=0\\\\\Rightarrow (x-a)(ax+1)=0
     \Rightarrow (x-a)=0  অথবা (xa+1)=0       

\Rightarrow x=a  অথবা  x=-\frac{1}{a} 
    \therefore x= 39  অথবা x= -\frac{1}{39} 

1. (x+7)(x+8)+\frac{21}{(22)^{2}} 


ধরি, 22=a\Rightarrow 21=a-1 
(x+7)(x+8)+\frac{21}{(22)^{2}}\\\\=x^{2}+15x+56+\frac{a-1}{a^{2}}\\\\=\frac{1}{a^{2}}(a^{2}x^{2}+15a^{2}x+56a^{2}+a-1)\\\\=\frac{1}{a^{2}}[a^{2}x^{2}+15a^{2}x+(56a^{2}+a-1)]\\\\=\frac{1}{a^{2}}[a^{2}x^{2}+15a^{2}x+(56a^{2}+8a-7a-1)]\\\\=\frac{1}{a^{2}}[a^{2}x^{2}+15a^{2}x+8a(7a+1)-1(7a+1)]\\\\=\frac{1}{a^{2}}[a^{2}x^{2}+15a^{2}x+(8a-1)(7a+1)]\\\\=\frac{1}{a^{2}}[a^{2}x^{2}+\left \{ a(8a-1)+a(7a+1) \right \}x+(8a-1)(7a+1)]\\\\=\frac{1}{a^{2}}[a^{2}x^{2}+a(8a-1)x+a(7a+1)x +(8a-1)(7a+1)]\\\\=\frac{1}{a^{2}}[ax(ax+8a-1)+(7a+1)(ax+8a-1)]\\\\=\frac{1}{a^{2}}[(ax+8a-1)(ax+7a+1)]\\\\=(x+8-\frac{1}{a})(x+7+\frac{1}{a})\\\\= (x+8-\frac{1}{22})(x+7+\frac{1}{22})

সমাধান করো নিজেরা : 

1.  x^{2}-2000\frac{2000}{2021}x-1=0 

2. x^{2}-65\frac{32}{33}x-2=0 

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