Final answer:
The maximum possible value of |x+y||x-y| is 3.
Step-by-step explanation:
To find the maximum value of |x+y||x-y|, we need to consider the relationship between the magnitudes of complex numbers. Let's express x as x = r1 * e^(iθ1) and y as y = r2 * e^(iθ2), where r1 = 1, r2 = 2, and θ1, θ2 are the arguments of x and y respectively. Then x+y = (r1*e^(iθ1)) + (r2*e^(iθ2)), and x-y = (r1*e^(iθ1)) - (r2*e^(iθ2)). We can express |x+y||x-y| as |(r1*e^(iθ1)) + (r2*e^(iθ2))|| (r1*e^(iθ1)) - (r2*e^(iθ2))|. Utilizing the properties of complex conjugates and the modulus function, we can simplify this expression to |x+y||x-y| = |r1^2 - r2^2 + 2 r1 r2 cos(θ1 - θ2)|. Since r1 = 1, r2 = 2, and the maximum value of cos(θ1 - θ2) is 1, the maximum possible value of |x+y||x-y| is |1^2 - 2^2 + 2(1)(2)(1)| = 3.