**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.