With those two equations and the constraint, you have three equations and three unknowns.
ah yep.
so from first one λ = 1 - 1/x
sub that into the second one gives y = -2x
sub that into the constraint gives 5x^2 = 5 => x^2 = +1 or -1
for x=-1
y= +2 or -2
and for x= 1
y= +2 or -2.
so extreme points at (-1,-2), (-1, 2), (1,-2), (1, 2). Correct?