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Falqueza, Glyndel Kate A.GE-1123
Math 10408-14-15
A. Solvn! equaton"
4;4
4
16
16
1601616
016
21
2
2
2
2
−==
±=
=
=
+=+−
=−
x x
x
x
x
x
x
1.
[ ]
2
3;
2
3
2
3
2
132
2
1
94
94
90994
094
21
2
2
2
2
−==
±=
±=
=
=
+=+−
=−
t t
t
t
t
t
t
t
2.
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8/17/2019 Mathematics Equation
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[ ]
3
6;
3
6
3
6
3
3
3
2
3
2
3
123
3
1
23
)23(12)23(3
123
21
2
2
2
2222
22
−==
±=
•=
=
−−=−−
−=−
−−++=−−+−
+=−
x x
x
x
x
x
x
x x x x
x x
3.
[ ]
it it
it
i x
i x
x
x
x
2
3;
2
3
2
3
2
132
2
1
32
94
90994
094
21
2
2
2
−==
±=
±=
±=
−=
−=−+
=+
4.
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8/17/2019 Mathematics Equation
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5.
3;3
3
9
9
)1(8)1(12
17912
)3(3
17)3)(3(312)3(
3
17)3)(3(
3
12
3
173
3
12
21
2
2
2222
22
2
2
2
−==
±=
=
=
+−++=+−+−
+−=−
−
−+−+=
−−−
−
+−+=
−
−
−++=
−
−
x x
x
x
x
x x x x
x x
x x x x
x x x
x
x x
x
x
x x
x
x
#. Solvn! $y %a&to'n!.
4
4044
0)4(
0
0)(
0)4(
04
)4(4)4(
4
2
1
2
2
2
=+=+−
=−
==
=−
=−
−+=−+
=
t
t
t
or
t
t
t t
t t
t t t t
t t
1.
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8/17/2019 Mathematics Equation
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4
4044
0)4(
7
7077
0)7(
0)4)(7(
0283
2828283
283
2
1
2
2
2
=+=+−
=−
−=−=−+
=+=−+
=−+
−=−+
=+
x
x
x
or
x
x
x
x x
x x
x x
x x
2.
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8/17/2019 Mathematics Equation
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[ ]
2
2022
0)2(
2
1
2
112
2
1
10112
0)12(
0)2)(12(
0252
22252
225
2
1
2
2
2
=
+=+−=−
=
−−=−−
−=−+−=+−
=−+−=−+−
−=−+−
=−
x
x
x
or
x
x
x
x
x x
x x
x x
x x
3.
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[ ]
5
5055
0)5(
5
8
5
185
5
1
80885
0)85(
0)5)(85(
040335
)208(20255)208(208)45(5)52(4
4)4)(1(4
5
)4)(1(
)1()4(4)4)(1(
4
5
4
1
1
1
2
1
2
2
2
=+=+−
=−
=
=
+=+−=−
=−−=+−
+−++−=+−+−+−=−
−−
=
−−−+−
−−
=−
+−
t
t
t
or
t
t
t
t
t t
t t
t t t t t t t t
t t t t
t t t t
t t
4.
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[ ]
[ ]
2
3
2
132
2
1
32
0)32(
3
2
3
123
3
1
0)23(
0)32)(23(
0656
656
)2)(56(56
3
2)2)(56(
56
3
2
2
1
22
22
p y
p y
p y
p y
or
p y
p y
p y
p y p y
p py y
p py y
p p y p y
p
p
y p p y
p y
p
p
y
=
=
==−
−=
−=
=+
=−+
=−−=−
−
−=−
−=
5.
(. Solvn! $y &o)*letn! the "qua'e and quad'at&
equaton.
1. $y quad'at& %o')ula+ $y &o)*letn! the "qua'e+
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2
3133
4
3133
13
1)3(
1)3(
9896
086
2
1
2
2
2
2
=+−=+−
=+=+−
±=−
=−
=−
+−=+−
=+−
x
x
x
x
x
x
x
x x
x x
2
2
26
4
2
26
2
26
2
32366
)1(2
)8)(1(4)6(6
2
4
086
2
2
1
1
2
2
2
=
−=
=
+=
±=
−±=
−−±
=
−±−=
=+−
x
x
x
x
x
x
x
a
acbb x
x x
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4
2
3
2
5
1
2
3
2
5
2
5
2
3
4
25
2
3
4
94
4
93
34
2
2
1
1
2
2
2
−=
−−=
=
−=
±=+
=
+
+=++
−=
t
t
t
t
t
t
t t
t t
4
2
53
1
2
53
2
53
2
1693
)1(2
)4)(1(4)3(3
2
4
043
34
2
2
1
1
2
2
2
2
−=
−−=
=
+−=
±−=
+±−=
−−−±−=
−±−=
=−+
−=
x
x
x
x
x
x
x
a
acbb x
t t
t t
2. $y quad'at& %o')ula+
$y &o)*letn! the "qua'e+
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[ ]
3
5
6
281
6
28
6
28
6
8
3
1
6
8
6
8
9
1
6
89
16
3
5
9
16
3
8
3
10583
3
1
0583
2
2
1
1
2
2
2
2
−=
−−=−=
+−=
±−=
−±=−+
=
+
+−=++
=++
=++
x
x
x
x
x
x
x
x
x x
x x
3
5
6
281
6
28
6
28
6
60648
)3(2
)5)(3(4882
4
0583
2
2
1
1
2
2
2
−=
−−=−=
+−=
±−=
−±−=
−±−=
−±−=
=++
x
x
x
x
x
x
x
a
acbb x
x x
3. $y quad'at& %o')ula+
$y &o)*letn! the "qua'e+
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4. $y quad'at& %o')ula+ $y &o)*letn! the
"qua'e+
32
2
324
32
2
324
2
324
2
124
2
4164
)1(2
)1)(1(444
2
4
014
2
2
1
1
2
2
2
+=
−−=
+−=
+−=
±−=
±−=
−±−=
−±−=
−±−=
=++
x
x
x
x
x
x
x
x
a
acbb x
x x
( )
32
23
32
23
32
32
4144
014
2
2
1
1
2
2
2
+=
+=
+−=
−=±=+
=+
+−=++
=++
x
x
x
x
x
x
x x
x x
5. $y quad'at& %o')ula+ $y &o)*letn! the
"qua'e+
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8/17/2019 Mathematics Equation
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[ ]
i x
i x
i x
x
x x
x x
x x
x x
54
53
5
4
5
3
5
4
5
3
5
3
5
3
25
16
5
3
25
91
25
9
5
6
01
5
6
5
10565
5
1
0565
2
1
2
2
2
2
2
−=
+=
±=+−
−=
−
+−=+−
=+−
=+−
=+−
i x
i x
i x
x
x
a
acbb x
x x
54
53
5
4
5
3
10
86
10
646
)5(2
)5)(5(4)6()6(2
4
0565
2
1
2
2
2
−=
+=
±=
−±=
−−±−−=
−±−=
=+−
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8/17/2019 Mathematics Equation
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.52
)48(4
)4)(3(42
4
0423
2
2
2
=
−−=−−=
−=
=−+
D
D
D
acb D
y y
ete')nn! the &ha'a&te' o% the 'oot".
1.
D>0 and is not a perfect square therefore the roots are real,
irrational andunequal.
31
321
)4)(2(4)1(
4
042
2
2
2
−=−=
−−=
−=
=+−
D
D
D
acb D
x x
2.
D
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8/17/2019 Mathematics Equation
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D0 and is a perfect square therefore the roots are real,
rational andunequal.
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8/17/2019 Mathematics Equation
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E. Fndn! quad'at& equaton" th nte!e' &oe&ent".
023
0221
0)2)(1(
2,1
2
2
=+−=+−−
=−−==
x x
x x x
x x
x x
1.
06
0623
0)2)(3(
2,3
2
2
=−+
=−−+
=−+=−=
x x
x x x
x x
x x
2.
0121115
15015
12
15
1115
015
12
5
3
3
4
05
3
3
4
5
3
,3
4
2
2
2
=−−
=−−
=−+−
=
+
−
−==
x x
x x
x x x
x x
x x
3.
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8/17/2019 Mathematics Equation
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( )
04
44444
444
)2(2
2222
22
22,22
2
2
2
22
=−
−=−+−
=+−
±=−
−±=−
±=
−=+=
x x
x x
x x
x
x
x
x x
4.
( ) ( )
0)311()6(
023693)2(3
0)3)(23()3()23(
0)3()23(
3,23
2
2
2
=−+−+
=++−+−+−+
=+−+++−+
=++−+
−−=+−=
i xi x
iiix x xi x x
ii xi xi x
i xi x
i xi x
5.
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8/17/2019 Mathematics Equation
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F. Solvn! the value o% &on"tant *.
[ ]
3
22
3
1223
3
1
223
0223
04318
04)3()3(2
3
042
2
2
−=
−=
−==+
=++=+−−−
−==+−
p
p
p
p
p
p
x
px x
1.
[ ]
14
144
4
1
426
426
)2(33
2
2
2)2(3
3
2
025)2( 2
=
=
=−+=
+
=
++
=
=+++
p
p
p p
p p
p p
p p
a
c
p x x p
2.
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8/17/2019 Mathematics Equation
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[ ]
2
5
1105
5
1
6438
4836
)3)(12(3
4
)12(
)2()3)(12(
017)2()12(
3
4
1722
2
22
=
=
+=−+−=−−
−
−=
−+−−
=−−++−
−=
−++=++
p
p
p p
p p
p p
p p
p x p x p
a
b
p x x px px
3.
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G. Solvn! equaton".
( ) ( )
[ ]
( )0
0
8
108
8
1
81616
416
416
216
216
22
22
22
==
=
++=+
+=+=−+
=
−+
=−+
x
x
x
x x x
x x
x x
x x
x x
1.
( )
1
01
2
02
0)1)(2(
02
86
26
26
026
2
1
2
2
33
3 2
3 2
3 2
−==+
==−
=+−=−−
=+−
=+−
=+−
=−+−
x
x
or
x
x
x x
x x
x x
x x
x x
x x
2.
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8/17/2019 Mathematics Equation
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362 −− x x
3. let y/
( ) ( )
133;133
13)3(
9496
046
136
136
:
2
1
0)2)(1(
02
05)3(
5366
21
2
2
2
2
22
2
2
2
22
−=+=
±=−
+=+−
=−−
=−−
−=−−
=
−==−+
=−−
=−−+
=−−−−
x x
x
x x
x x
x x
x x
substitute
y
or
y
y y
y y
y y
x x x x
( ) ( )
7
07
1
01
0)7)(1(
076
436
236
:
4
3
2
2
22
2
==−
−==+
=−+=−−
=−−
=−−
x
x
or
x
x
x x
x x
x x
x x
substitute
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8/17/2019 Mathematics Equation
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8/17/2019 Mathematics Equation
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( )
213109282
21
3109282
42
31094164
42
49744164
)21(2
)272)(21(4)164(164
2
4
027216421
0324008200425
43284
252004004
3282
5)4(
2
)328(5
5
2
328
4
2
)328(5
5
2
328
4
2
1
2
2
2
22
22
22
2
2
2
2
2
−=
+=
±=
±=
−−±=
−±−=
=+−
=−++−−
+−=
+−
+−=
−
+−
=
+−
−+−
=+−
−
x
x
x
x
x
a
acbb x
x x
x x x x
x x x x
x x x
x x
x x
x x x
x x
x
4.
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8/17/2019 Mathematics Equation
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.
2 x y =Solvn! equaton".
1;1
1
1
1
43
2
2
−==±==
=
x x
x
x
x
3;3
3
9
9
:
1
01
9
09
0)1)(9(
0910
0910
21
2
2
2
24
−==±==
=
==−
==−
=−−=+−
=+−
x x
x
x
x
substitute
y
y
or
y
y
y y
y y
x x
1. let
2 x y =
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8/17/2019 Mathematics Equation
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i xi x
i x
x
x
−==±=
−=
−=
43
2
2
;
1
1
2
2;
2
2
2
2
2
1
2
1
:
1
01
2
1
012
0)1)(12(
012
012
21
2
2
2
24
−==
±=
=
=
−==+
=
=−=+−
=−+
=−+
x x
x
x
x
substitute
y
y
or
y
y
y y
y y
x x
2. let
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8/17/2019 Mathematics Equation
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4 x y =3. let
( ) ( )
( ) ( )16
2
2
11
1
:
2
02
1
010)2)(1(
02
02
2
444
4
1
444
4
2
4
==
=
=−=
−=
==−
−==+
=−+
=−−
=−−
x
x
x
or
x x
x
substitute
y
y
or
y
y y y
y y
x x
x x y 1+=
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8/17/2019 Mathematics Equation
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202
2
1
012
0)2)(12(
0252
522
251
4
3
2
2
==−
=
=−=−−
=+−
=+
=+
x x
or
x
x
x x
x x
x x
x x
2
3
2
1;
2
3
2
1
2
31
)1(2
)1)(1)(4(11
2
4
01
1
11
:
1
01
2
5
052
0)1)(52(
0572
051
71
2
21
2
2
2
2
2
2
i x
i x
x
x
a
acbb x
x x
x x
x x
substitute
y
y
or
y
y
y y
y y
x x
x x
−=+=
−±=
−±=
−±−=
=+−=+
=+
==−
=
=−
=−−=+−
=+
+−
+
4. let
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8/17/2019 Mathematics Equation
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( ) 22 −+= x y5. let
( )
( )
( )
( )
( )
( )[ ]
222;
222
2
22
2
11222
2
1
2221
)2(22
1)2(
22
1
22
1
22
43
2
2
2
i xi x
i x
ii xi
i
i xi
xi x
x
x
x
x
−−=+−=
±−=
+−=
+=
+
=
++
−=+
−=+
−=+ −
( ) ( )
( )
( )
( )
5;132
23
)3)(2(3
1
2
1)3)(2(
9
1
2
1
9
1
2
1
9
1
2
:
2
02
9
1
0190)2)(19(
02179
0221729
21
2
2
2
2
24
−== ±−=
+=±
±+
±=
+±+
=+
=+
=+
−==+
=
=−=+−
=−+
=−+++
−
−−
x x x
x
x x
x
x
x
x
substitute
y
y
or
y
y y y
y y
x x
