Each of m and n is a complex number that satisfies this set of simultaneous equations:

• mn - n² = 3
• mn + 2m² = 1

Without solving for m and n, determine the value of mn

Note: mn is the product of m and n, rather than their concatenations.

(1): mn - n^2 = 3
(2): mn + 2m^2 = 1
(3): Multiply (1)&(2)
(4): Add (1)&(2)

(3): (mn - n^2)*(mn + 2m^2) = 3
(mn)^2 + mn(2m^2 - n^2) - 2*(mn)^2 - 3 = 0
-(mn)^2 + mn(2m^2 - n^2)  - 3 = 0
+(mn)^2 - mn(2m^2 - n^2)  + 3 = 0

(4): 2mn + 2m^2 - n^2 = 4
2m^2 - n^2 = 4 - 2mn

(mn)^2 - mn(4 - 2mn)  + 3 = 0
Let x = mn
x^2 - 4x + 2x^2 + 3 = 0
3x^2 - 4x + 3 = 0

x = mn = (2 + sqrt(5)*i)/3  or  (2 - sqrt(5)*i)/3

Posted by Larry on 2023-01-08 10:07:55