Problem Set 1
PROBLEMS
1.) The sum of two numbers is and there difference is
. What are the two numbers?
2.) Find the values of and
if:
3.) Prove that
4.) Define and
a.) Prove that
b.) Prove that
SOLUTIONS AND PROOFS
Posted October 13, 2009
1.) Solution: Let and
be the two numbers. Then,
and
. Adding the equations, we have
. Substituting it to the first equation gives us
. Therefore, the two numbers are
and
.
2.) Solution: From the given, and
are roots of of the cubic equation
Factoring, we have
Therefore,
or
3.) Proof: We know that the square of the difference of any two numbers is always positive or . Let
be any two numbers. Then,
. Expanding, we have
. Adding
to both sides of the equation yields
. Getting the square root of both, we have,
4.) Proof (a): We want so we will just replace
‘s with
. Therefore,
Proof of 4b is left as an exercise. It’s very similar to the proof of 4a.