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.