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