If x = √(a+2b)+√(a-2b)/√(a+2b)-√(a-... then prove that bx^2 - ax + b=0.
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Answered by
165
x ={ √ ( a + 2b) + √( a -2b)} /{ √( a+2b)-√(a-2b) }
take componendo and devedendo rule
(x +1)/( x -1) = {√(a +2b) +√(a -2b)+√(a +2b)-√(a -2b)}/{ √(a +2b) +√(a -2b)-√(a +2b) +√( a -2b)}
(x +1)/(x -1) = √(a +2b)/√(a -2b)
take square both sides
(x +1)²/(x -1)² = (a +2b)/(a -2b)
(a -2b)(x +1)² = (a +2b)(x -1)²
(a -2b)x² +2(a -2b)x +(a -2b) = (a +2b)x²-2(a +2b)x +( a +2b)
( a +2b -a +2b)x² -2x(a +2b+a-2b)x +( a+2b-a +2b)
4bx² -4ax + 4b = 0
bx² -ax + b =0
hence proved ////
take componendo and devedendo rule
(x +1)/( x -1) = {√(a +2b) +√(a -2b)+√(a +2b)-√(a -2b)}/{ √(a +2b) +√(a -2b)-√(a +2b) +√( a -2b)}
(x +1)/(x -1) = √(a +2b)/√(a -2b)
take square both sides
(x +1)²/(x -1)² = (a +2b)/(a -2b)
(a -2b)(x +1)² = (a +2b)(x -1)²
(a -2b)x² +2(a -2b)x +(a -2b) = (a +2b)x²-2(a +2b)x +( a +2b)
( a +2b -a +2b)x² -2x(a +2b+a-2b)x +( a+2b-a +2b)
4bx² -4ax + 4b = 0
bx² -ax + b =0
hence proved ////
Lipimishra2:
Thanks a lot for helping out. The question itself is a bit too difficult for me, so it took quite some time figuring out the processes and steps you wrote. As I was not quite familiar with the componendo, divendo method. But I understand now. Thanks a lot again for the help!
One doubt tho. Most friends whom I asked about it, first tried rationalizing the denominator out. (Then they thought of expanding it or something, but couldn't think more. )Can that be a method too?
yeah , there are many method to solving this ,
but if you know about componendo and dividendo rule thrn this type of question easily solve
so, i think you should be read this rule , if any help join my chat i will help you
Answered by
34
Hope this will help you...
x ={ √ ( a + 2b) + √( a -2b)} /{ √( a+2b)-√(a-2b) }
take componendo and devedendo rule
(x +1)/( x -1) = {√(a +2b) +√(a -2b)+√(a +2b)-√(a -2b)}/{ √(a +2b) +√(a -2b)-√(a +2b) +√( a -2b)}
(x +1)/(x -1) = √(a +2b)/√(a -2b)
take square both sides
(x +1)²/(x -1)² = (a +2b)/(a -2b)
(a -2b)(x +1)² = (a +2b)(x -1)²
(a -2b)x² +2(a -2b)x +(a -2b) = (a +2b)x²-2(a +2b)x +( a +2b)
( a +2b -a +2b)x² -2x(a +2b+a-2b)x +( a+2b-a +2b)
4bx² -4ax + 4b = 0
bx² -ax + b =0
hence proved ....
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