Two numbers are such that the sum of their cubes is 5 and the sum of their squares is 3. Find the sum of the numbers.
If a and b are integers, then there are a few possibilities :
3-ab=1 and a+b=5 (1)
3-ab=5 and a+b=1 (2)
3-ab=-1 and a+b=-5 (3)
3-ab=-5 and a+b=-1 (4)
From (1), we get ab=2, that is a=1 and b=2 for example. But a+b is not equal to 5. Impossible for integers.
From (2), we get ab=-2. We want a+b=1. --> a=-1 and b=2.
From (3), we get ab=4 --> a=-1 and b=-4
From (4), we get ab=8. Impossible for integers.
Alternatively, let the numbers be and . we have
So as Moo said,
Now, let and , then we have the system
Solve this system for and you have your answer (since is the sum of the two numbers)
I like your solution, Moo
But, of course, who said they had to be integers?