1.

If `x=(sqrt(p^6+q^2)+sqrt(p^2-q^2))/(sqrt(p^2+q^2)-sqrt(p^2-q^2))` then `q^2x^2-2p^2x+q^2=?`

Answer» `x = (sqrt(p^2+ q^2) + sqrt(p^2 - q^2))/(sqrt (p^2 + q^2) - sqrt(p^2 - q^2))`
multiply by `sqrt(p^2 + q^2) + sqrt(p^2 - q^2)`
`x = ((sqrt(p^2 + q^2) + sqrt(p^2 - q^2))^2)/(sqrt(p^2 + q^2) - sqrt(p^2 - q^2))`
`= (p^2 + q^2 + p^2 - q^2 + 2 sqrt((p^2+ q^2)(p^2- q^2)))/(p^2 + q^2 - p^2 + q^2)`
`x = (p^2 + sqrt(p^4 - q^4))/q^2`
`x^2 = (p^4 + p^4 - q^4 + 2 p^2 sqrt(p^4 - q^4))/q^4`
`= (2p^4 - q^4 + 2p^2 sqrt(p^4 - q^4))/q^2 - (2p^4 - 2p^2 sqrt(p^4 - q^4))/q^2 + q^2`
`= -q^4/q^2 + q^2 = 0`
here, `p=1, q=1`
`x = (sqrt2 - 0)/(sqrt2 - 0) = 1`
`q^2x^2 - 2p^2x + q^2`
`1 xx 1 - 2 + 1 = 0`
answer


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