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the roots of the equation (a+sqrt(b))^(x...

the roots of the equation `(a+sqrt(b))^(x^2-15)+(a-sqrt(b))^(x^2-15)=2a` where `a^2-b=1` are

A

`+-2,+-sqrt(3)`

B

`+-4,+-sqrt(14)`

C

`+-3,+-sqrt(5)`

D

`+-6,+-sqrt(20)`

Text Solution

Verified by Experts

The correct Answer is:
B
We have `(a+sqrt(b))(a-sqrt(b))=a^(2)-b=1`[given]
`:.(a+sqrt(b))^(x^(2)-15)+(a-sqrt(b))^(x^(2)-15)=2a`
`implies(a+sqrt(b))^(x^(2)-15)+1/((a+sqrt(b))^(x^(2)-15))=2a`
Let `y=(a+sqrt(b))^(x^(2)-15)`
`impliesy+1/y=2aimpliesy^(2)-2ay+1=0`
`impliesy+(2a+-sqrt(4a^(2)-4))/2=a+-sqrt(a^(2)-1)`
`:.y=a+-sqrt(b)=(a+sqrt(b))^(+-1)[:'a^(2)-b=1]`
`implies(a+sqrt(b)^(x^(2)-15)=(a+sqrt(b))^(+-1)`
`:.x^(2)-15=+-1`
`impliesx^(2)=15+-1impliesx^(2)=16,14`
`impliesx=+-4,+-sqrt(14)`
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