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
+53 Hello Guest,
(a) Let's start by squaring. We have \(a^2+b^2+2ab=16.\) Thus \(a^2+b^2+2ab=a^4+b^4.\)We soon see that the only solutions for \(\boxed{ab=0.}\)
(b) Problem explictly states that \(a+b=4\).
(c) We know that \(a+b=4\) and \(ab=0\). Thus one of either \(a\) or \(b\) must be zero.
There are four solutions: \((a,b)=(2,0);(-2,0);(0,2);(0,-2).\)
