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Complex Equation Conclusion (Posted on 2015-12-12) Difficulty: 3 of 5
A and B are complex numbers such that:

A2 + B2 = 7
A3 + B3 = 10

What is the maximum possible real value of A + B?

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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Solution Solution Comment 3 of 3 |
Consider the complex numbers by real and imaginary parts.
A=a+ci
B=b+di


A2 = (a2 - c2) +(2ac)i
B2 = (b2 - d2) +(2bd)i
A3 = (a3 - 3ac2) + (3a2c - c3)i 
A3 = (b3 - 3bd2) + (3b2d - d3)i 


The system to solve is then
(a2 - c2) + (b2 - d2) = 7
(2ac) + (2bd) = 0
(a3 - 3ac2) + (b3 - 3bd2) =10
(3a2c - c3+ (3b2d - d3) = 0

If A+B is real we also know c=-d *see below
and with the second equation of the system we also get a=b 

Substitute these into the first and third equations to get the simpler system:
2a2 - 2c2 = 7
2a3 - 6ac2 = 10 

Solving for a (multiply the top by 3a and subtract the bottom)
gives the cubic 4a3 - 21a + 10 = 0
which has roots -2.5, .5, 0

The complex number solutions to the original system are then 
A = -2.5 + isqrt(11)/2, B = -2.5 - isqrt(11)/2, A+B=-5

A = .5 - sqrt(13)/2, B = .5 + sqrt(13)/3, A+B=1
(note they are real)
A = 2 + isqrt(1/2), B = 2 = isqrt(1/2), A+B=4

*I didn't know what to make of "real value of A+B" so I took it as a hint.  These solutions seem to work but the above does not rule out other solutions where A+B is complex but their real parts sum to more than 4.

  Posted by Jer on 2015-12-14 11:07:40
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