Let x, y be real numbers such that x y=1 . Let T and t be the largest and the smallest values of the expression (x+y)^2-(x-y)-2 (x+y)^2+(x-y)-2 If T+t can be expressed in the form m n where m, n are nonzero integers with GCD (m, n)=1 , find

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Let x, y be real numbers such that x y=1 . Let T and t be the largest and the smallest values of the expression (x+y)^2-(x-y)-2 (x+y)^2+(x-y)-2 If T+t can be expressed in the form m n where m, n are nonzero integers with GCD (m, n)=1 , find the value of m+n .

Quantrex Academy · Accepted Answer

Correct answer: 25

aligned & x y=1 (x+y)^2=(x-y)^2+4 & (x+y)^2-(x-y)-2 (x+y)^2+(x-y)-2 = (x-y)^2-(x-y)+2 (x-y)^2+(x-y)+2 & = (x- 1 x )^2- (x- 1 x )+2 (x- 1 x )^2+ (x- 1 x )+2 & x- 1 x R & Let f(t)= t^2-t+2 t^2+t+2 , t R aligned aligned & f(t)_ +f(t)_ = 4+ 2 4- 2 + 4- 2 4+ 2 = 18 7 = m n & m+n=25 aligned

Let x, y be real numbers such that x y=1 . Let T and t be the largest and the smallest values of the expression (x+y)^2-(x-y)-2 (x+y)^2+(x-y)-2 If T+t can be expressed in the form m n where m, n are nonzero integers with GCD (m, n)=1 , find the value of m+n .

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