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Complex Equation Conclusion (Posted on 2015-12-12)

A and B are complex numbers such that:

A² + B² = 7
A³ + B³ = 10

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

See The Solution Submitted by K Sengupta

Rating: 4.5000 (2 votes)

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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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