Discrete Math ProofsAre these steps for finding the solutions of x + 3 = 3...
Discrete Math Proofs
Are these steps for finding the solutions of correct?
1. is given;
2. , obtained by squaring both sides of (1);
3. , obtained by subtracting from both sides of (2);
4. , obtained by factoring the right-hand side of (3);
5. or , which follows from (4) because implies that or .
Answer & Explanation
Those steps are correct, but there's one final step required:
6. When , the left-hand side is 2 and the right-hand side is 2; when , the left-hand side is 3 and the right-hand side is -3. So only the first "solution" is a solution. (Er. When I did this the first time, I wrongly thought that the RHS in the case was 3.)
To explain step 5: we want to find x, given that . Let and ; then since , we have or . So or ; so or .
Why is it the case that if then or ? This is precisely the statement that the product of nonzero quantities is nonzero (by taking the contrapositive), which you might find more intuitive; but I see there is another answer which gives algebraic manipulations to prove it.
Yes, all the steps are correct. You only need to check those value of x if it satisfies the original equation or not.
Assume that . We want to show that either or .
If then we are done. Suppose that . Then . Thus, using field properties of R we get