If `alpha,beta` are roots of `x^2+-p x+1=0a n dgamma,delta` are the roots of `x^2+q x+1=0` , then prove that `q^2-p^2=(alpha-gamma)(beta-gamma)(alpha+delta)(beta+delta)` .

Explanation

Since `gamma, delta` are roots of `x^(2) + qx + 1 = 0,` we have
`gamma^(2) + qgamma + 1 = 0 or gamma^(2) + 1 = - qgamma` (1)
and `delta^(2) + 1 = - qdelta` (2)
Now , `(alpha - gamma ) (beta - gamma)(alpha + delta)(beta + delta)`

= `{alphabeta = gamma(alpha + beta)+gamma^(2)}{(alphabeta+ delta(alpha + beta)+ delta^(2)}`
`(1 + pgamma + gamma^(2)) (1 - pdelta + delta^(2))`
=`(pgamma = qgamma) (-pdelta - qdelta)` [from (1) and (2)]
= `-gammadelta (p-q) (p+q) = q^(2) - p^(2)`