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Solve for (5+2sqrt(6))^(x^2-3)+(5-2sqrt(...

Solve for `(5+2sqrt(6))^(x^2-3)+(5-2sqrt(6))^(x^2-3)=10` (1985,5M)

Text Solution

Verified by Experts

The correct Answer is:
`+-2,pmsqrt2`
Given, `(5+sqrt6)^(x^(2)-3)implies(5-2sqrt6)^(x^(2)-3)=10" "...(i)`
Put `y=(5+2sqrt6)^(x^(2)-3)implies(5-2sqrt6)^(x^(2)-3)=1/y`
From Eq. (i), `y+1/y=10`
` y^(2)-10y+1=0impliesy=5+-2sqrt6`
implies (5+2sqrt6)^(x^(2)-3)=5+2sqrt6`
`or (5+2sqrt6)^(x^(2)-3)=5+2sqrt6`
`impliesx^(2)-3=1or x^(2)-3=-1`
`x=+-2 or x=+-sqrt2`
`implies x=+-2, +-sqrt2`
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