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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 2 of 3 |
Start with the identites:
(A+B)^2 = (A^2+B^2) + 2(A*B)
(A+B)^3 = (A^3+B^3) + 3*(A+B)*(A*B)

Rearrange the first as:
A*B = [(A+B)^2 - (A^2+B^2)]/2

Substitute into the second:
(A+B)^3 = (A^3+B^3) + 3*(A+B)*[(A+B)^2 - (A^2+B^2)]/2

Let A+B=S for convenience and plug in the given values to get:
S^3 = 10 + 3*S*[S^2 - 7]/2
S^3 = 10 + (3/2)*S^3 - (21/2)*S
0 = S^3 - 21S + 20
S = 4, 1, or -5

Then the maximum value of A+B is 4

  Posted by Brian Smith on 2015-12-12 21:26:43
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