epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

Collection of Proofs 117
Alleged Proof 2.
We need to solve the equation:
4 4*
If we write all the terms in the left-hand side, we get:
So,
4 4
_ 5/4 lb V25/16 - 1 _ 5/4 ± 3/4
2 2
The two solutions are x = 1 and x = 1/4.
Alleged Proof 3. A second-degree equation has at most two solutions.
So, the graphs meet in at most two points. •
Alleged Proof 4. To show that the statement is true, one needs to show
that the two graphs have in common no points, one point, or two points. To
find the coordinates of the points one must solve the equation:
•
4 4*
By factoring, the equation can be rewritten as:
By dividing both sides, one gets:
4^-l) = 4(:x;-1).
1
" = 4-
Therefore, the graphs of the two equations have in common only the point
with coordinates (1/4, -3/16). Thus, the given statement is true. •
THEOREM 5. l£t n be an integer with n > 1. Then n^ — n is always even.
Alleged Proof 1.
We will use proof by induction.
Step 1: Is the statement true when n= 1?
1^-1=0