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Let a and b be real numbers such that a + b = 4 and a^4 + b^4 = 16.

(a) Find all possible values of ab
(b) Find all possible values of a + b
(c) Find all possible values of a and b

 Dec 2, 2022
 #1
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Attention Melody, please:

 

If you square a + b =4, you get this:


a^2 + b^2 + 2ab =16


Set ab=p and note that: 


a^2 + b^2=2[8 - p]. Square again to get:


a^4 + 2a^2b^2 + b^4 =4[8 - p]^2


And therefore, the right-hand side:


=16 + 2p^2=4(p^2 -16p + 64). Simplify and re-order, you get:


P^2 - 32p + 120=0


p==16+or-sqrt(132)


Melody: I have reached this far! But, I'm stuck as how to get the values of a and b ?

 Dec 2, 2022
 #2
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There are NO real solutions for a and b. There are several "complex" solutions, however.

 Dec 4, 2022

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