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8
32. (a) [BB] False:
x
=
y
=
0 is a counterexample.
(b) False:
a
=
6 is a counterexample.
(c) [BB] False:
x
=
0 is a counterexample.
(d) False:
a
=
Vi,
b
=
Vi
is a counterexample.
(e) [BB] False:
a
=
b
=
Vi
is a counterexample.
Solutions to Exercises
b± Vb
2
4ac
(f)
The roots of the polynomial
ax
2
+
bx
+
c are
x
=
2a
.
If
b
2

4ac
>
0,
.../b
2

4ac
b+ Vb
2
4ac
is real and not 0, so the formula produces two distinct real numbers
x
=
2a
and
b

.../b
2

4ac
x=
2a
(g) False:
x
=
~
is a counterexample.
(h) True:
If
n
is a positive integer, then
n
~
1, so
n
2
=
n(n)
~
n.
33. The result is false. A square and a rectangle (which is not a square) have equal angles but not pairwise
proportional sides.
34. (a) [BB] Sincen
2
+1 is even,
n
2
is odd, so
n
must also be odd. Writingn
=
2k+l,
then
n
2
+1
=
2m
says
4k2
+
4k
+ 2
=
2m,
so m
=
2k2
+
2k
+ 1
=
(k
+ 1)2 +
k
2
is the sum of two squares as
required.
(b) [BB] We are given thatn
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.
 Summer '10
 any
 Graph Theory

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