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Thus the equation

z

2

2
±
(
az
)+

a

2
)+

a

2
=
ρ
2
, becomes
(
x

u
)
2
+(
y

v
)
2
=
ρ
2
,
which is the equation for a circle of radius
ρ
centered at (
u, v
), which
in complex notation is the point
a
=
u
+
iv
.
5. We have

z

2
=(
±
z
)
2
+(
²
z
)
2
≥
(
±
z
)
2
,
and taking square roots gives

z

≥

±
z

.
Similarly for

z

≥

²
z

.
We have

z
+
w

2
=(
z
+
w
)
(
z
+
w
)
=
z
z
+
z
w
+
w
z
+
w
w
=

z

2
+2
±
(
z
w
)+

w

2
.
Putting these equations together we have

z
+
w

2
=

z

2
+2
±
(
z
w
)+

w

2
≤

z

2
+2

¯
zw

+

w

2
=

z

2
+2

zw

+

w

2
=(

z

+

w

)
2
.
7. Recall that a circle in
R
2
centered at (
a, b
) with radius
r
is given by
the equation
(
x

a
)
2
+(
y

b
)
2
=
r
2
.
We will manipulate the equation
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This note was uploaded on 07/27/2010 for the course MATH Math2009 taught by Professor Koskesh during the Spring '09 term at SUNY Empire State.
 Spring '09
 Koskesh
 Math, Square Roots

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