Master Math: Algebra Debra
CAREER PRESS 3 Tice Road P.O. Box 687 Franklin Lakes, NJ 07417 1 -800-CAREER-1 20 1-848-03 ...
436 downloads
2765 Views
4MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
Master Math: Algebra Debra
CAREER PRESS 3 Tice Road P.O. Box 687 Franklin Lakes, NJ 07417 1 -800-CAREER-1 20 1-848-03 10 (NJ outside U.S.) 201-848-1727
0 1996 by
book by by
MASTER MATH: ALGEBRA by by
VISA
on
Library of Congress Cataloging-in-PublicationData
/ by
:
p. )
1. 1996 5
Acknowledgments
Dr.
This page intentionally left blank
aaaaaaaaaamamaoamaaaaaammaooaaamaamaaaaaaaaoaaaaaaaaaaaaaaaa
Table of Contents
aaamaaaaaaammmoamaaaaaaamaaamaaamammamaaaaamamaaaaaaaaaamama
Introduction
1
Chapter 1 Translating Problems into Algebraic Equations
5
1.1. 1.2.
5
1.3. 1.4.
6 8 12
Chapter 2 Simplifying Algebraic Equations
22
2.1. 22 2.2. 2.3.
23 26
2.4.
by
28 2.5. 30
Chapter 3 Solving Simple Algebraic Equations
31
3.1. 31 3.2. 39 3.3. 43
Chapter 4 Algebraic Inequalities
46
4.1. 46
Chapter 5 Polynomials
49
5.1. 5.2. 5.3. 5.4. 5.5. 5.6.
49 51 52 53 55 61
5.7. 62
Chapter 6 Algebraic Fractions with Polynomial Expressions 6.1. 6.2. 6.3. 6.4.
72 72 73 74 76
Chapter 7 Solving Quadratic Polynomial Equations with One Unknown Variable
81
7.1 82
7.2 83 7.3 87 7.4 90
7.5. 92
Chapter 8 Solving Systems of Linear Equations with Two or Three Unknown Variables 8.1.
95
of
96 8.2.
98
8.3. 103 8.4.
105 8.5. 110 8.6.
111 8.7.
115 8.8.
119 8.9.
126
Chapter 9 Working with Coordinate Systems and Graphing Equations 9.1. 9.2. 9.3. 9.4. 9.5. 9.6. 9.7.
133 134 139 145 150 152 155 160
9.8. on
164
Index
169
Appendix Tables of Contents of First and Third Books in the Mwter Math Series
172
This page intentionally left blank
Introduction
Algebra ter Math Basic Math and Re-Algebra metry. Master Math
MasPre-Calculus and Geo-
book
Algebra
Algebra
book
Algebra
Algebra
1
Algebra
book on on book
Master Math
Algebra
Master
Math no
Muster Math book
A
2
Introduction
“I do
up.
3
This page intentionally left blank
Chapter 1
Translating Problems intoAlgebraic Equations 1.1 1.2 1.3 1.4
1.1. Introduction to algebra
you you
$1.00 you
5
$20.00 you
buy
Algebra
“X.”
$1.00 $20.00?
you $20.00. $20.00.
x
$1.00 by $1.00
=
1.2. Translating English into algebraic equations
by added to, plus, sum, in all, altogether, increased by, the sum of, more, total. by difference,how many less, subtracted from, how many more, reduced by, decreased by, minus, less, take away. 6
Translating Problems into Algebraic Equations
by multiplied by, of, times, product, multiply, twice by 2), double by 2 ) . by divided by, quotient, divides, ratio. 2nd
squared
cubed
by equals, is equal, is, are, were, was, the sum is, the difference is, the quotient is, the product is, the same as, results in, the result is, makes, leaves, yields, gives. by a number, what.
38
8
an unknown, a variable,
3 x
3x + 8 = 38. 11.
5 y
(2y)2 - 5 = 11.
5 n
of 5 + n3 = 13. 7
13.
Algebra
1.3. Algebra terminology
A
Letters by
b,
A,
x, y, z,
A variable
by x, y, z,
A d,
b,
n,
constants. by or by
coeficient. 3y,
3
y -4
8
x
Translating Problems into Algebraic Equations
=3 * n
3n =
2*3#23
As expression
numerical
x4y
algebraic expression.
An equation
up
An
9
Algebra
+, -
=
by
+
terms. no 3x
2x
3
2.2.
9x, 6
2x, 2x.
by
Inequalities
on on >, <, 2
I,
.
10
Translating Problems into Algebraic Equations
factors
= 18x,
x
x
=
x2,
x
18
x
Polynomials no “1”
4x4 - 2x2 + 4 45x2 3x + 9 -8
+
5.1.
Linear equations do
1. 2x
+ 1= 5 x
A,
Ax y
+
=C ,
y= b y
on
9 on
11
+ b,
Algebra
Non-Linear equations 1. 1= 5
1.4. Simple word problems
Important note: 2,
3,
Perimeter Word Problems
1, book Master Math and Geometry
Be-CaZcuZus 2
3 2
4
+3
+4
=9
4
6 2(4
+
+ 12
=8
12
= 20
Translating Problems into Algebraic Equations
Area Word Problems
1, book Master Math CaZcuZus and Geometry
Pre-
=
=
3 2
(3
=6
Volume Word Problems
book Pre-CaZcuZus and Geometry
=
13
1, Master Math
Algebra
3
2
4 = 24
(3
Price and Profit Word Problems
=
- (%
=
$24
$24 = $X - (0.25)($~) x= $24 = $x(l $24 = $x(0.75) = $X
$x = $32 $32 i s 2
=
3
-
14
Translating Problems into Algebraic Equations
$16, gross
-
=
$S - $16 = $S - $16 = $S - $16 = $8 $S = $8 $16 $S = $24
+-
$24. Percent Word Problems o
(
17 +
20
= 85%
by 100,
by
$350.00
(40%)/(
= 0.40
15
Algebra
by = $140
$140. $350 - $140 = $210
Distance Word Problems
=
60
30
2 2
4
60
2 = 120
30
2 = 60
4 120
+ 60
= 180 =
up 16
Translating Problems into Algebraic Equations
Work Word Problems
1/ n
+ 1 / m = 1/ h ,
n do h do
1
by n
do Mary
1,000 1,000
4 6 1,000
1/4+ 1/6= lk
A
12.
+
=
=l k
by =1
17
Algebra
by h= h = 1 2 6 = 2.4 2.4
Mixture Word Problems
you $1.20 pound pound,
6 2
$1.60 pound
+
$1 =
$7.20 + $3.20 = $10.40 =
6
1.20
2
by 8 =
18
Translating Problems into Algebraic Equations
15 pound 15 pound
y 15 =
(15 =
by
+
+
+y = + 3 + y = 5.25 + 0 . 3 5 ~ y-
+y
15
+
= 5.25 -
2
3
0 . 6 5= ~ 2.25 y = 2.2510.65 y = 3.46 3.46
so
oat
19
3
Algebra
Interest Word Problems simple interest
x
=
x
$4,000
3% 3
1
3%
3/100 12
(0.03)($4,000)(1
= $120
12
3 ($120)(3 mo./12
= $30
$4,000 + $30 = $4,030 =
3 3
compound interest, by by 4, by 2,
20
Translating Problems into Algebraic Equations
$5,000
5%
1 5%
6 = $125
$5,000
+
6 = $5,125
6 5%
1
on = $128.12
6
$5,125 + 1
= $5,253.12
21
Chapter 2
Simplifying Algebraic Equations 2.1 2.2 2.3 2.4
by
2.5
2.1. Commutative, associative and distributive properties of addition and multiplication
22
Simplifying Algebraic Equations
commutative
b
associative
+ + x
+ +
= (b x = x x
b
distributive x
+
=
x
+
x
+c)= + b
2.2. Using associative and distributive properties The
Using the Associative Property
23
Algebra
+ +3=6
+ + 1= 6 3+ + 2
+ +
1 =6
+ + + = 10 + 2 + + 4 = 10 3 + +4)+ 1 = 10 4 + +2 + 10 + + + = 10 2 x 3 x 5 = 30 3x 2 x 5 5 x 3x 2 5 x 2 x 3 = 30
+ + = 1x3 + 1x4 + 2x3 + 2x4 =3 + + 1+ = 3x1 + 3x2 + 4x1 + 4x2 = 3 + 6 + 4 + 8 =21 5.4,
24
Simplifying Algebraic Equations
Using the Distributive Property
+
2(3 + 4) = (2)(3) (2)(4) = 6 + 8 = 14 2(4 - 3) = (2)(4) - (2)(3) = 8 - 6 2 =I
for
on
undistributing unfactoring. 2(3
+
+
= (2)(3) (2)(5)
2
+
(2)(3)+ (2)(5)= 2(3 5) 2
2(3 + (2)(3)+ (2)(2)
25
Algebra
2.3. Combining like terms in algebraic equations
Like terms
b, c, d, x,y, z, x2,
&,&, G,
b2, c3, d4, x2b,
do 2x2, x2
26
2
Simplifying Algebraic Equations
3-h
2b
& 2b
2b2 1 0 6
no =
‘‘1” xl/n
=
&
only
27
Algebra
-
-
=
+
0 =
+
-
+
=
2.4. Simplifying algebraic equations by removing parentheses and combining like terms by
3y -
3y - 2y + 3y - 2y - 6 = 6
-
=6
-
28
+
= 6.
Simplifying Algebraic Equations
y by
6
"="
)
3y - 2(y 36 36 -
+
+
=6
=6
+ 2
+ +
+ -
=4
=4
=4 =4
29
-
= 4.
Algebra
+
-
= 2.
no
+
-
+
-
-
=2
=2
=2
2.5. The general order to perform operations in algebra
by
30
"+"
"-"
Chanter 3
Solving Simple Algebraic Equations 3.1
3.2 3.3
3.1. Solving algebraic equations that have one unknown variable
on
=
31
Algebra
y
y
y=3+2 y=5
on
y, y + 3 - 3 ~- 63 y+0=3 y=3
+ 3 = 6.
y
y
z
42 = 32.
3
by y 3 = 6.
+
y = 3.
3+3=6 6=6
by 4.
z,
4214 = 3214
2=8 32
So 1v ing Simple Algebraic Equations
by 42 = 32. z = 8.
(4)(8) = 32 32 = 32
- 6 = 25. - 6 +6 =25 +6 a+O=31 a=31
6
by
- 6 = 25. = 31.
31 - 6 ~ 2 5 25 = 25 x x,
x/4 = 2. by 4.
(x/4)(4)= (2)(4) (4x/4) =8
33
Algebra
by x/4 = 2. x = 8. 814 = 2 2=2
x
+
x,
4
x/4 4 - 4 = 2 - 4 x/4+0=-2 x/4 = -2
by 4. (d4)(4)= (-2)(4) (4x/4)= =
x=
34
x/4
+ 4 = 2.
Solving Simple Algebraic Equations
by
+
x/4 + 4 = 2.
-8/4 4 = 2 -2+4=2 2=2 z z,
-Z
- 4 +4 =2
-z- 4 = 2.
4
+4
- ~ = 6
by -1. -z,
z,
z = lz
-z = lz.)
(-z)/( - 1) = (6)/(- 1) = 1,
z = -6
by -z - 4 = 2.
-(-6) - 4 = 2
x
=
6-4=2
by
by
35
Algebra
y 8y - 3y
8y
+ 5 - 3y = 10.
by
+ 5 = 10 8y - 3y =
- 3)y = 5y,
5
5y
+ 5 - 5 = 10 - 5 by 5. = 515
by
8y
+5 -
+ 5 - 3y = 10. = 10
13 - 3 = 10 10 = 10
on
by by
36
or
Solving Simple Algebraic Equations
+
3x - 10 = -4 x.
x
by -x
x
3x - x - 10 = 2x - 10 = -4 10 2x - 10 10 = -4+ 10
+
by 2. = 612
by 3x - 10 = -4+ x. - 10 = -4+ 3 9-10=-4+3 -1= -1
by by x 10 +
- 20 =
10
+
-
-
by
5x 37
+
= -5(4
Algebra
+
10 + 10 + - 20 = 10 - 20 + =
+
+
-
=
10 + 10 +
+ 10
=
=
by 11. =
x=
by
+
10 + 10
-
+
10 +
+
-
10 x=
- 20 =
= -5(4
+ = -5(4 + = -5(4
+
- 10 = + 20 - 10 + 20 = -20 + 20 + + 10 = 60111 + 60111 + 10 =
10 = 10 = 10 = 10
+
38
Solving Simple Algebraic Equations
3.2. Solving simple algebraic equations containing fractions
by
by
1. 2. by
3.
4. by
5.
2 d 3 - 2d4 = 4
2x-2x- - = 3
4
lx -2x- - = 3 2
4 1 4 1 39
x.
Algebra
3 2
3 , 6 , 9 , 12, ... 2 , 4 , 6 , 8 , ...
6
by -(6)2x - - 3
(6)lx - -(6)4 2 1
( 2 ) 2 -~(3)lx= (6)4
4x - 3x = lx. l x = 24 x=24
by 2(24)/3 - 2(24)/4 = 4 4813 - 2412 = 411 6.
by 6.
40
Sol u ing Simple Algebraic Equations
-(2)48 - - 1
(3)24 - - 4 1 1
24 = 24 = Zd(2xi-4)
(2x+4), 2x -6 -10 2(x+2)
by
- x (5)(x+ 2) 5
(x+ 2)
+ 2)= x(5) 3x + 6 = 5x 3x
41
x.
Algebra
by 2.
612 = X 3=x
by
6 10
(x = 3)
6 6+4
6 - -6 10
10
on
6 _.
10
2x 2x+4 (6)*(2x+4)
(10)*(2x).
(6)(2x+4)= (10)(2x)
12x+24=20x 12x 24 = 2 0 -~1 2 ~ 42
Solving Simple Algebraic Equations
24 = 8x by 8.
24/8 = x 3 =x - = -
b
= bx.
y
3.3. Solving simple algebraic equations containing radicals
1.
on
2.
3. 4.
by
2
43
Algebra
2 d z = 8
on
by 2.
d Z = =
8 2
-
4
=
42
(dKT)(dX-l) =
16
1
by = 8
2 d E 2&
=
8
44
Solving Simple Algebraic Equations
by 2.
f i =8 2
16 = 16
&
=
4x4=
4
=
(x1/2)2= x
(G3=
(x113)3= x
(G4=
(x1/4)4= x
45
Chapter 4
Algebraic Inequalities
4.1
4.1. Solving algebraic inequalities with one unknown variable >,
by
<, 2
1.19., Basic Math and Pre-Algebra,
5.
book
by
Inequalities (>) (2)
(<),
or
on on
46
Algebraic Inequalities
x 3 -(x/3) s - 5 - 3 -(x/3) <
by -3.
x 2 (-8)(-3) x 2 24 47
3 - ( d 3 )< -5.
Algebra
by 3-
3-
I
5 8-
s0
8
by 3. 24 I x
48
ChaDter 5
Polynomials
5.1 5.2 5.3
5.4 5.5 5.6 5.7 ax2
+ bx + c
5.1. Definitions
A monomiaZ
2x, 4 x 5 8
binomid 2~ + 8 , 3 - ~5x2
49
Algebra
A trinomiaZ 4x2 + 3x + 9, 4x4 - 2x2 + 4x 2x2
degree
2.
degree by 4x4 - 2x2 + 4x 5x2 + 3x + 9 4x2
4. 2. 2. 2.
0.
8
2x
1.
“1”
=
2
50
Polynomials
5.2. Addition of polynomials by
(y2
- 3y
+ 4)
+
(y - 3yz y3).
(y2-3y+4)+(y-3y2+y3) =y3+y2-3y2-3y+y+4 =y3
(y2
- 2y2 - 2y + 4
- 3y + 4) + (y - 3y2+y3) =y3 - 2y2 - 2y + 4 2y(y2 - 2y
2y(y2 -
+ 2)
3(y - 4y2
+ 2~3).
zY+ 2) + 3(y - 4y2 + 2y3). 2y
3
+ 4y + 3y - 1 2 ~ +2 6y3 = 2y3 + 6y3 - 4y2 - 12y2+ 4y + 3y
= 2y3 - 4y2
Add = By3 -
16y2
2yk2 - 2y
+ 7y
+ 2) + 3k - 4y2 + 2y3) = By3 - 16y2 + 7y 51
Algebra
5.3. Subtraction of polynomials by
- 3y +
(y - 3y2+
by
- 3y + 4 - y + 3y2 - y3
= =
+ 3y2 - 3y - y + 4 + 4y2 - 4y + 4 - 3y +4) - (y -
=
52
- 4y + 4
Polynomials
5.4. Multiplication of polynomials
muZtipZy monomials,
=
=
=
=
=
multiply a monomial with apolynomial,
+
-
= =
+
=
-
+ + 60x2 + 20x
= 24x
+ 53
+ 48
Algebra
multiply two binomials,
First, Outer, Inner
Last
multiply polynomials with polynomials,
on.
54
POlynomials
+ 2x - 5) = + + + + + = 2x3 + 4x2 + + + + 20 = 2x3 + 4x2 + + + + 20 = 2x3 - 18x + 20 (2x -
5.5. Division of polynomials by by
divide monomials,
+
- 6x =-
-3x2
2 2 6x - - - - --3xx -X
divide polynomials by monomials,
55
Algebra
(4x +
+
+
+6 =
+
=
+
= 4d6
2 3
+
= -x
- 6x -
-
-
+
+
= --
2x
+ -6x + 2x
4 -
3
+
-
=
+
-
+ 2 - -- -x 3 2x
+ + l/x
divide polynomials by polynomials, do by
no
56
Polynomials
+ +
(-1Ox 8 8x2) -+ (2x - 4):
Example:
?
8x2 + 2x = 4x. 4x Z X -) 4 82 ~ -10~+8 4x(2x - 4) = 8x2 - 16x 4x J
2 x - 4 8x2 - l O x + 8
8x2 -16x 4x Z X - 4J 8~ 2 - 1 0 +~ 8 8x2 - 16x 6x+8 by
6x, 6x +- 2x = 3.
57
2x
Algebra
+ 3
4x )
2
2 ~ - 8~ 4 OX+ 8 8x2 - 16x 6x+8 3(2x - 4) = 6x - 12
4x
+ 3
2x - 4 8x2 -1Ox + 8 8x2 - 16x 6x+ 8 6~ - 12
no
4x
+ 3
2~ - 4) 8~ 2 - 1 0 +~8 8x2 - 16x 6x+ 8 6~ - 12 20 20 (-IOx+ 8 + 8x2) + (2x - 4) = 4x + 3 + 2x- 4 20 20 - - 10 -2 ~ - 4 Z(X-2) x-2 4x + 3 + (lO/(x-2)).
final
58
Polynomials
+
+
(x2 4) +- (x 4):
?
x+4’x2+0x+4 x2 + x = x. X
x + 4 ’ x 2 +Ox+4
+
+
x(x 4) = x2 4x X
x + 4 ) x 2+ o x + 4 x2+ 4x
X
x + 4 ) x 2+ O x + 4 x2+ 4x - 4x+4
by
-4x,-4x + x = -4. 59
x
Algebra
x - 4 x+4)x2+0x+4 x 2 +4x - 4x+4
-4(x+ 4) = -4x x -4 x + 4Ix2 + ox + 4 x2 +4x -4x +4 - 4 - 16 ~
no
x -4 x+4’x2+ Ox + 4 x2+ 4x - 4x+4 - 4 ~1620
20 (x2 + 4) + (x + 4) = x - 4 + x+4’
60
Polynomials
5.6. Factoring polynomials with a common monomial factor
6= 2
6.
3
2a + 2b = 2(a + 2
+
2x + 5x = x 2x2 + 4x2y = 2x2
+
= 7x
1+
12x4 - 6x3 3x2 = 3x2
- 2x + 61
Algebra
5.7. Factoring polynomial expressions with the form ax2+bx+c ax2
ax2
(x + 2)(x + 3).
+
+
(x 2)(x + 3) = x2 + 3x 2x + (2)(3)
+ ( 2 + 3 ) +~ (2)(3)= x2 + 5x + 6 (4x + 2)(5x + 3). (4x + 2)(5x + 3)
= x2
+ (4x)(3)+ (2)(5x)+ (2)(3)
= (4x)(5x) = 20x2
+ 12x + 10x + 6 = 20x2 + 22x + 6
62
+ bx + c,
+ bx + c
Polynomials
Compare the factored binomial form toith the trinomial form n
+
+ = x2
= x2
+
+ nx +
+ + bx +
p, q,
+
= 1, b =
= pqx2
+ pnx +
+
+
ax2
b=
=
n
+
+ = pqx2
+
=
63
+ bx +
=qp,
Algebra
1.
(
)(
).
2.
3. 4.
.)
5.
x2
(
x
only
)(
+ 5x + 6.
).
x.
1
(x
2
1: ( (
3
3) 6) 64
1
6
Polynomials
5x.
+ 5x.
3x + 2x = 5x 2: 6x + -1x = 5x no 2
(x + 2)(x + 3),
1
(x + 2)(x + 3) = x2 + 3x+ ZX+ 6 = x2+ 5~ + 6 x2
(x + 2)(x + 3).
+ 5x + 6
x2 + 3x - 18. (
x
)(
).
x.
(x >(x )
6 2,18
-3, -6 1.
65
3,9
Algebra
3x.
+ 3x. (x + 6)(x - 3) = 3x (x - 6)(x + 3) = -3x (x + 9)(x - 2) = 7x (x - 9)(x + 2) = -7x (x 18)(x - 1) = 17x (x - 18)(x+ 1) = -17x
= -3x,
= 6x,
= 3x,
= -6x,
= -2x,
= 9x,
= ZX,
+
3x
66
= -9x,
= -lx,
= 18x,
= lx,
= -18x,
(x + 6)(x - 3).
Polynomials (X
+ 6 ) ( -~3) = X'
(X
+ 6 ) ( -~3).
+ 3x - 18 x2 + 3x - 18
- 3~ + 6~ - 18 = X'
20x2 + 22x (
)(
+ 6.
).
2x lOx, 20x (2x )(10x ) (20x )(lx ) (5x )(4x 1
lx, 5x
4x.
2 ( (
3,l 6. 2)( 3) I)( 6)
22x. 22x. (2x ) ( l O X ) (20x >(lx ) (5x )(4x 1 ( 2)( 3) ( 6)
67
Algebra
+ 20x,
+
2x,
= 6x,
=
= 26x
+
= 60x,
=
= 62x
+ 8x,
+
+
lOx,
+
+ 4x,
+
= 15x,
=
= 23x
+
=
=
= 22x
+
= 120x,
=
= 121x
+
= 30x,
=
= 34x
no
22x
+
+
+
+
= 20x2
+
+
+ 12x + 10x + 6 = 20x2 + 22x + 6 20x2 + 22x + 6
68
Polynomials
Special Binomial Products to Remember
+
differenceof two squares, x2 - y2, -
+
-
= x2
-
= x2
- xy + xy - y2 = x2 - y2 - 3x + 3x - 32
- 32
-9
4x2 - 25. = 4x2
25?
(5)(5)= 25 4x2 - 25 =
+ 5)(2x - 5).
by
+ = 4x2 +
-
+ - 25 = 4x2 +
- 25
= 4x2 -
- 25 = 4x2 - 25
+
sum of two squares, x2 y2, binomiaZ squared
+ 2xy + y2 + = x2 + xy + xy + y2 =x2 + 2xy + y2 = x2 - 2xy + y2 - y)(x - = x2 - xy - xy + y2 = x2 - 2xy + y2
+ +
= x2
69
Algebra
x4 - 81.
~4
- 81 = (x' + 9 ) ( ~ -29) (x2 - 9)
+
x2 - 9 = (x 3)(x - 3) x4 - 81
+
(x2 + 9)(x 3)(x - 3).
+
+
(x2 9)(x 3)(x - 3). (x 3)(x - 3) = x2 - 3x 3x - 32 = x2 - 9 (x2 9)(x2 - 9)= x4 - 9x2 9x2 - 92 = x4 - 81
+ +
ax2
+
+
+ bx + c 2x2
+ 1Ox + 12. 2.
+
2x2 + 1 0 ~12 = 2(x2 + 5x + 6)
+ + 6).
(x2 5x
70
Polynomials
no
+ +
(x + 2)(x 3) (x + 6)(x 1)
)
(x + 2)(x+ 3) = 5x (x + 6)(x+ 1) = 7x
= 3x,
= 2x,
= lx,
= 6x,
5x,
(x + 2)(x + 3). 2
(2)(x+ 2)(x + 3).
+
+ + +
(2)(x+ 2)(x 3) = (2)(x2 3x 2x 6) = (2)(x2+ 5x + 6) = 2x2 + 1Ox + 12
2x2 + 1Ox + 12
(2)(x+ 2)(x + 3).
71
Chapter 6
Algebraic Fractions with Polynomial Expressions 6.1 6.2 6.3 6.4
6.1. Factoring and reducing algebraic fractions
by
72
Algebraic Fractions with Polynomial Expressions
--
-
-
-
+ 1
=
-
=
+
-
+ 5x + x2 x2
+ 4x +
-
--
-
6.2. Multiplication of algebraic fractions
by
by
73
Algebra
+
+
(x2
+ 6x + x2
+
+ 5x +
24x2 + 6x +
6x
(x+ 6x
-
+
-
+ 2)
24x2 + +
+ +
+
+
-
- 4x2 + 8 x -
(x+
6.3. Division of algebraic fractions
by by
74
Algebraic Fractions with Polynomial Expressions
by by
by
(x' + 4~ + 4)/(2~2 - 8) + (x' + 2 ~ ) / ( -48)~ x 2 + 4 x + 4 -. ( x 2 + 2 x ) 2x2 - 8 4 ~8 -
up
by
24)( ~ - 2 ) - x 2 + 4 x + 4X 4 x - 8 - ( ~ + 2 ) ( ~ + X 2x2 - 8 (x2+2x) 2(x2- 4 ) x ( x +2)
- ( x + 2 ) ( x + 2 )X 4 ( x - 2 ) - (x + 2) 2(x+2)(x-2) x(x+2) 2 ( x - 2 )
+
(x2 - 4) = (x 2)(x - 2).
75
X
4(x - 2 ) x(x+2)
Algebra
6.4. Addition and subtraction of algebraic fractions
Fractions with Common Denominators
by
2x/(3(x - 2)) - 4/(3(x - 2)).
Fractions With Different Denominators
1.
76
Algebraic Fractions with Polynomial Expressions
if
+
you
6, 6
4 12, 16...,
4 4, 8, 6, 12, 24, 30...,
12.
2.
by
=
3.
by
77
Algebra
+
(x2 + 4x + 4)/(2x2 - 8) + (4x - 8)/(x2 2x).
x2+4x+4 + 4 x 4 -- ( X + 2) ( X + 2) + 4 ( x - 2) 2x2 - 8 x2+2x 2(x+2)(x-2) x(x+2) - ( ~ + 2 ) 4(~-2) +
2(x-2)
(2)(x- 2)
x(x+2)
+
(x)(x 2). (2)(x)(x 2)(x - 2).
+
by
(x+ 2) 2(x-2)
+
(x)(x 2). x(x + 2) x ( x + 2) 78
Algebraic Fractions with Polynomial Expressions
4(x - 2) x(x+2) (2)(x - 2). 2(x - 2 ) 2(x-2)
by
( X + 2) (x)( X + 2)
+ (4) ( X - 2) (2)( X - 2)
79
Algebra
x3 + 12x2 - 28x + 32 - x3 + 12x2 - 28x + 32 2x3 - 8x (2x)(x2- 4 )
+
(x2 + 4x + 4)/(2x2 - 8) + (4x - 8)/(x2 2x) x 3 + 12x2 - 28x + 32 2x3 - 8x
80
Chapter 7
Solving Quadratic Polynomial Equations with One UnknownVariable 7.1 7.2
7.3 7.4
7.5
81
Algebra
7.1. Defining and solving quadratic (polynomial) equations
Quadratic equations x 2
x2. ax2
a
b by
b
+
bx c = 0 3x2 + 2x + 5 = 0 3x2 + 2x = 0
82
+ bx +
= 0.
Quadratic Polynomial Equations, One Unknown Variable
1. 2.
3. b zero,
4. b
7.2. Using factoring to solve quadratic equations with one unknown variable
:
1. ax2
+ bx + c = 0. 83
Algebra
+ bx +
2.
3. by
4.
x2 - 2x = 3.
+ bx + -
- 3=0
= 0.
+ bx +
- 2x -
+
-
+
-
+
=
-
=
+
= 3x,
= 2x
+
84
= =
Quadratic Polynomial Equations, One Unknown Variable
by ( x + l)(x - 3) = x 2 - 3x+ l x - 3 =x2 - 2x - 3 (x + (x - 3). (x+ 1 ) = 0 (x - 3) = 0 x+1=0 x=-1 ~ - 3 ~ x=3
0 x2 - 2x = 3,
x=-landx=3. by x2 - 2x = 3 For x = -1, (-1)2- 2(-1)= 3 1+2=3 3=3 x = 3, 32 - 2(3) = 3 9-6=3 3=3
85
Algebra
x2 = 2x.
+ bx +
= 0.
+ bx + -
=0
x
x2 = 2x, by x2 = 2x x = 0, 02 = x = 2, 22 = 4=4
86
x =0
x = 2.
Quadratic Polynomial Equations, One Unknown Variable
7.3. Using the quadratic formula to solve quadratic equations with one unknown variable quadratic formula
discriminant, b2 b2 b2 -
1.
> 0, < 0, = 0,
2 2 1
+ bx + c = 0.
2.
b 87
Algebra
b
3.
by
4.
. by
5.
3x2 + 2 =
+ bx +
3x2
+ 5x + 2 = 0
= 0.
b b
k by
k
J25-24 6
k
fi
6 88
Quadratic Polynomial Equations, One Unknown Variable
so, 4
+
f = 1.
-5 + l x = 6
-
-5 - 1 x = 6
3x2 + 2 = -5x x = -2/3
x = -1.
by 3x2 2 = -5x. x = -2/3, 3(-2/3)2 2 = -5(-2/3) 3(4/9) 2 = 10/3 12/9 2 = 10/3 4/3 + 2 = 10/3
+
+ + +
2/1= 6/3,
4/3 + 613 = 10/3 10/3 = 10/3 For x = -1, 3(-1)2+ 2 = - 5(-1) 3(1) + 2 = 5 5=5
89
Algebra
7.4. Using the square root method to solve quadratic equations with one unknown variable square root method b b ax2
+ bx +
=0
ax2+c=o.
1.
x2
on
2
x2
x,
3. by
4.
3x2 + 4 = 31. 4
3x2 = 27 x2 by 3.
90
dx2 = k x.
Quadratic Polynomial Equations, One Unknown Variable x2 = 2713 x2 = 9
x = +3 3x2 + 4 = 31
x=3
x = -3. by 3x2 + 4 = 31.
x = 3, 3(3)2 4 = 31 3(9) + 4 = 31 27 + 4 = 31 31 = 31 x = -3, 3(-3)2 4 = 31 3(9) 4 = 31 27+4=31 31 = 31
+
+ +
91
Algebra
7.5. Using the method of completing the square to solve quadratic equations with one unknown variable method of completing the square
b
x2
1.
by
2.
+ bx = + =
by b,
b,
3.
by
4.
x2
5.
on
by
92
Quadratic Polynomial Equations, One Unknown Variable
x2 - 2x - 10
=
-7.
x2
+ bx =
10 =
3 by b,
= (-2/2)2 =
b, = 1.
= =4
=4
-
-
= 4 =
-
-
-
=
= 4
= 4
x2
-
on
=4
x2
by
93
-
Algebra
x-l=&2
@ = kx . x
+2
-2.
+2, x-l=2 x=3 -2, x - l = -2 x = -1 x2
x=3
x=
by x2 - 2x - 10 = x = 3, (3)' - 2(3) - 10 = -7 7 9 - 6 - 3= 0
x= (-1)2 - 2(-1)- 10
=
-7 7
1+2-3= 0
o=o 94
- 2x 10 = 7
Chanter 8
Solving Systems of Linear Equations with Two or Three Unknown Variables 8.1
8.2 8.3 8.4
8.5 8.6
8.7
8.8
95
Algebra
8.9
8.1. Solving systerils of linear equations with two or more unknown variables
n
n
96
Solving Systems of Linear Equations
Setting up word problems using two variables
20, 10? x
y x + y = 20.
20,
x - y = 10.
10,
50
you you
10% 14% 14% 50
10%
x y 14% 50
x
+y
= 50
(8%)(x
(10%)(50 are
97
+ (14%)(y
=
Algebra
Methods for two equations and two unknowns
Methods for three equations and three unknowns
8.2. Using the elimination method to solve systems of linear equations with two unknown variables e h i n a t i o n method by by by
98
Solving Systems of Linear Equations
1.
2. by
3.
4.
by
5.
x - y =4
x
x y
x
+
y 3x = 4x
+ 2y = y 99
Algebra
x by
y x - y = 4 by 2 =
=
y 8
- 2y = 8
+ 2y 5x + 0
=
7
= 15
5x = 15 x = x =3 y. 4 = 4
= = =
y
1
=
x y =4 x =3
y=
100
3x + 2y = 7
Solving Systems of Linear Equations
by x - y =4 x-y = 4 3= 4 3+1 = 4 4 = 4
3x = =
3x
+ 2y = 7
7
7
20
10. x y 20,
x + y = 20.
10,
x - y = 10.
x + y = 2 0 + x - y = 10 2x
= 30
101
+ 2y = 7.
Algebra
2x = 30 x = 3012 x = 15 x = 15 y. = 10
15 - y = 10 15 - 10 = y
x + y = 20 by =
x +y
=
20
20
20 = 20
x-y
=
15 - 5 = 10 10 = 10
102
10
x - y = 10
Solving Systems of Linear Equations
8.3. Using the substitution method to solve systems of linear equations with two unknown variables substitution method by
1. by on 2.
3. 4.
5.
by
103
on
Algebra
2x - y = 4
2x + 4y = 4.
by
on
2x-y = 4 2 ~ - =4 y y
2x+4y = 4 2 ~ + 4 ( 2 ~ -=4 4)
x 16
4
ZX + 4 ( 2 -~4)
=
4
2 ~ + 8 16 ~ = - 4 1ox = 20 by 10. x = 20/10 x = 2
x =2 y.
2x-y = 4 2(2) - y = 4 4-y = 4 4
y
4-4
104
Solving Systems of Linear Equations
2x - y = 4
2x + 4y = 4
by 2x - y = 4
-0
= 4
2x = =
+ 4y = 4
4
4
8.4. Using the method of determinants to solve systems of two linear equations with two unknown variables method of determinants by
105
Algebra
A matrix.
2 by 2
determinant
3
4
-
2 by 2
(1)(4)- (3)(2)= 4 - 6 = -2
method ofdeterminants Cramer’s ruZe x
106
Solving Systems of Linear Equations
1.
x+ y= x + b2 y = x
b
y
(D, D,
2.
Dy),
up
Also, 3.
x
y,
4.
x
107
y
Algebra
50
you you
50 x y
50
x
+y
= 50
+
=
(10%)(50
x
+
+
y=
x b2 y =
+
=
by 100
8x
+ 14y = 500 1,
1,
50, b2
8, 14
500, 108
Solving Systems of Linear Equations
x=IMD
b2
I
1 1
D= 8
Dx=
14
500
= (1)(14) - (8)(1) = 14 - 8 = 6
14
1 1 8
50 500 x
x=
= (50)(14)
- (500)(1) = 700 - 500 = 200
= (1)(500) - (8)(50) = 500 - 400 = 100
y
= 200/6 = 10013
y = Dy/D = 10016 = 50/3
109
Algebra
x
+
+ y = 50
= (0.1)(50)
y = 5013
x=
14%
by
+
= 50
= 50
50 = 50 (O.O8)(x)+
8/3
+
+
= (0.1)(50) =
=5
=5
5=5
8.5. Solving systems of three linear equations with three unknown variables
110
Solving Systems of Linear Equations
8.6. Using the elimination method to solve systems of three linear equations with three unknown variables ezimination method
111
Algebra
1. 2. by
3. 4.
5.
by
6.
x, y
+ 4x + +
3y 32 = 4 3y - 42 = 5 2y 52 = 4
x
112
Solving Systems of Linear Equations
+ 3y + 32 = 4 + 3y - 42 = 5
2x
+
x =-8
- 6y -
+
4
- 4y -
+
2x + 3y 32 = 4 6y - = by z.
6y -
= 9
+
= 4 = 13 = 13
by 10.
y.
y= y = 1.3 z by
y = 1.3
- z =9 113
Algebra
7.8 -
=I
9 9
=
x by
y = 1.3
2x + 3y + 32 = 4
+ + + 3.9 - 3.6 2x + 0.3 = 4
=4
4
0.3 2x = 3.7 by 2.
x = 1.85 2x
+ 3y + 32 = 4 + 3y - 42 = 5
x = 1.85, y = 1.3
=
by 2x + 3y + 32 = 4
+
+
3.7 + 3.9 - 3.6 = 4 7.6 - 3.6 = 4
=4
114
=
Solving Systems of Linear Equations
- 2+ ~3y - 42 = 5 -2(1.85)+ 3(1.3) - 4(-1.2) 5 -3.7+ 3.9+ 4.8= 5 0.2 + 4.8= 5 5=5 =I
4x + 2y + 5z = 4 4(1.85)+ Z(1.3)+ 5(-1.2) =4 7.4+ 2.6 - 6 = 4 10 - 6 = 4 4=4
8.7. Using the substitution method to solve systems of three linear equations with three unknown variables substitution method
two
115
Algebra
1.
on
on
2.
or
3.
by
+ 2y - 22 = 2~ - 6y + = 6 4x
y.
+
4x 2y - 22 = -2 2y = 22 - 4x - 2
by 2. y = (1/2)(2z - 4x - 2)
116
4x
2z
Solving Systems of Linear Equations
y
3x+2(z-2x-l)+z=-l 3x+ 22 - 4x - 2 z = -1
+
- x + 32 = -1 2 - ~ + 3 ~ = 1 y
x. 2~ - 6(z - 2~ - 1) + 6~ = 6 2~ - 6~ + 1 2 ~6 + 6~ = 6
+
14x + 6 = 6 6 14x = 0
by 14. x=o
z,
-x + 32 = 1. - ~ + 3 ~ = 1 -0+ 32 = 1 32=1 by 3. z = 1/3
x =0
117
Algebra
x =0
z=
y
-1 -
y= y= y= 4x
y.
-1
+ 2y - 22 = - 6y + 6~ 0, y = -
z=
by
4x
+ 2y - 22 = + -
=
=
= =
3x
+ 2y + z = + + + 113 =
=
=
-1= 118
Solving Systems of Linear Equations
+
2~ - 6y 62 = 6 2(0) - 6(-2/3)+ 6(1/3) = 6 12/3 613 = 6 4+2=6 6=6
+
8.8. Using the matrix method to solve systems of three linear equations with three unknown variables matrix method
on
on
119
Algebra
1.
by on
b
2.
by
3. by 4.
for
5.
by
+
x b2 y b3
+
z= = d2 z = d3
120
Solving Systems of Linear Equations
upper triangular matrix,
1 0
1
c2
d2
by
-
= -2
121
b3 b2
Algebra
1 1 1 - 1 2 - 1 1 0 -1 1 -1 - 2 1
1by -2
2 -2 + 2
0
1
+
-1 0
1 1 0 -3 0 2
-2 -1
-2
-3
-1
1
row 3
-2 x 2
1 . 0
1 1 -1 1 -1 - 2
2
2 2
1 2
row3
0
-1 2 0 - 3
1 -1
122
Solving Systems of Linear Equations
1 0
1
1
-
1
2
0
2
-1 2
+
0 0
2
0
-1
3.
0 -1
2
-1
-1 2
(-1)
1.
1 0 0
1 1
1-1 1 1 -1 2 b3
3
3. 0 3 3 + 0 -3 -1
0
1 0 0
0
1 1 0
2
3 2
3
5
3
1-1 1 1 2 5 123
2
Algebra
1,
c3
1 0 0
1 -1 1 1 1 5/2
1 1 0
+ ly + 12 = -1 ly+lz=l lz = 5/2 z = 5/2.
z = 5/2 ly+ lz= 1 y + 5/2 = 1 5/2
y = 1- (5/2) y=2/2 - 5/2 y = -312 y = -3/2
x + y + z = -1 x + (-3/2)+ (5/2)= -1 x + 2/2 = -1 x+l=-1 1 x = - 1 -1
z = 5/2
124
3 by 2.
Solving Systems of Linear Equations
x=
+ y - z = -2 x=
y=
z=
by
+ z= + + y
=
-2+ 1= =
2x z =o 2(-2)- (-312)+ 512 = 0 -4+ 812 = 0 -4+4=0
-x + y - z = -(-2) + (-3/2)- 512 = -2 2 - 812 = -2 2-4=-2 -2 =
125
Algebra
8.9. Using the method of determinants of a square matrix to solve systems of three linear equations with three unknown variables A square matrix
3 by 3
126
Solving Systems of Linear Equations
2
by 2
method of determinants of a square matrix, Cramer’s rule
1. x+ x b2 y + x + b3 y +
+
z= z = d2 z = d3
3 by 3
2.
Dy
D,),
b2 b3
127
Algebra
Dx=
Dy=
=
128
Solving Systems of Linear Equations
x, y
3.
z
x=DX/D y=DYD z=DJD by
4.
x, y
z
3 ~ + 2 ~ - 2 ~ = 8 - 6y 6~ = 2 1 0 ~8y 1 0 =~ -8
+
+ +
b, d 3 4 10
2 -6 8
-2 6 10
3(-60 - 48) - 4(20 - -16) + lO(12 - 12) = 3(-108)-
4(36) + lO(0) = -324 - 144 = -468 129
Algebra
8
2 -8
3 4 10
2 -6 8
8 2 -8
-2 6 10
-2 6 10
130
Solving Systems of Linear Equations
3 Dz= 4 10
2 -6
8 2
8
-8
+ lO(4 - -48) = 3(32) - 4(-80)+ lO(52) = 96 + 320 + 520 = 3(48 - 16) - 4(-16 - 64)
=
936
Calculate x, y and z. = -9361-468= 2 y = DylD = 4681-468 = -1 z = DzlD = 9361-468= -2
x=
Therefore the solutions to: 3 ~ + 2 ~ - 2 ~ = 8 4 ~ - 6 ~ + 6 ~ = 2 1 0 ~8y + 1 0 =~ -8 are x = 2, y = -1 and z = -2.
+
Check the results by substituting into the original equations. The first equation: 3x + 2y - 22 = 8 3(2) 2(-1) - 2(-2)= 8
+
131
Algebra
6 + -2 + 4 = 8 4+4=8
+ 6~= 2 4(2)- 6(-1)+ 6(-2)= 2 8 + 6 + -12=2 14 + -12= 2 4~ - 6y
IOX + 8y + 10 z = -8
lO(2)+ 8(-1)+ 10(-2) = -8 20 + -8+ -20= -8 12 + -20 = -8 -8 = -8
on
132
Chapter 9
Working with Coordinate Systems and Graphing Equations 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 on
133
Algebra
9.1. Introduction and definitions on on by
real number line. on coordinate on
by two-dimensional coordinate system. by three-dimensional coordinate system. on
on
by on
rectangular coordinate x
systems A x, y
cylindrical
spherical.
134
y,
Coordinate Systems and Graphing Equations
on
on on
Apoint on
-4
-3 -2 -1
0
1 2 3 4
A point on planar coordinate system
X axis
Y
on
on x = 1, y = 1.
135
Algebra
on
3on
x = 3, y = -3,
= 3.
on
-4 -3 -2 -1 0 1 2 3 4 -3
136
Coordinate Systems and Graphing Equations
on on 4
3 2
0
-3 -2 -1
'I
1 2
3
Xaxis
-31
Y
-3 on 2 on
on
(-3,
on
A A A A
x
x
on on
y y
137
Algebra
on on
+3 on
-3 on
+3 on
by
(x, y, z) = (3, -3,3).
A Line
by on on
axis
138
Coordinate Systems and Graphing Equations
Perpendicular Lines
(90")
ParaLZeL Lines do
9.2. Graphing linear equations
by on
A line
x
on
A Linear equation
139
by y
Algebra
A
by
1.
2. by
y
x,
x
y. by
3.
4. by x,
y
on
- 6y -
4~-
- 6y -
8.
+8 140
Coordinate Systems and Graphing Equations
- 4 - ~8y = 16 x = 0, -4(0) - 8y = 16 = 16 y = -16/8 -2
y.
y = 0, - 4 - ~8(0) = 16 - 4 =~16 x = -1614 x = -4
x.
-4.
4
3 2
by
x -4x - 8y = 16. on
x=1
y.
141
y,
Algebra - 8y = 16
-4 -
16 = 20
y = - 2018 = - 2 1/2 - 2 1/2>
An
Ax +
1.
on
=C
by
2.
x y. by
3.
by
4.
on
y =x
y.
x x = 1,
y.
y=3 3).
142
+ 2.
Coordinate Systems and Graphing Equations
by
y = x + 2.
3=1+2 3=3 x = -1, y=-1+2 y=l
y =x
by
2=0+2 2=2 x = 2,
y
y.
by 1=-1+2 1=1 x = 0, y=o+2 y=2
x
+ 2.
(-1,l). x y
y.
(0,2). x y y = x + 2.
y.
y=2+2 y=4
by
y = x + 2.
(2,4). x y
143
Algebra
x=
y.
by y =x
(1,3),
+ 2.
x
y
(0,2),
X axis
Y axis
b
To graph an inequality, on
144
(-2,O>,
Coordinate Systems and Graphing Equations
>
<,
on 2
I, on
on on on
9.3. Slope of a line
on
slope
by by by first
145
Algebra
second by
first second m. equation for the slope of a line
x2 - x1
4 3 2
(-4,0)
-2),
- y1 x2 - x1 XI,
y1, x2
y2.
Coordinate Systems and Graphing Equations
-2 - 0 0 - -4
-2 - - -1 4 2 -1/2.
= y2
-
x2 - x1
on
147
Algebra
- XO)
y - yo
(xo, on
(xo, yo), yo). (x
y - yo =m(x -
b
yo.
x x :
148
y,
y,
Coordinate Systems and Graphing Equations
find the equation of a line when the slope of the line and one point on the line are identified: y - yo = - xg) on
+
y - 4=
find the slope (m) and y-intercept (6) of a line when the equation known: 2x
+ 6y = 12, y=
y= y=
+
+2 y =m x b =2
=
+ b,
find the equation of a line when two points on the line are identified: - y1 x - x1 x2 - x1 _.
find the equation of a line when two intercept points (a, 0) on the X-axis and (0, b) on the Y-axis are identified:
149
Algebra
9.4. Graphs of the equations for the parabola
y =ax2
x=
+ bx + + by +
y=
150
+ bx + c ,
Coordinate Systems and Graphing Equations
horizontal-axis parabola
A
axis of symmetry
axip
x, =
x, = 2,
drawn 2 on x,
X-axis.
=2
x
y = a x 2 + bx + y (x,,
151
yv
Algebra
9.5. Graphing quadratic equations quadratic equation
x, =
+ bx, +
x,
yv =
yv
x on y
y,).
x,
An x y
2x2 - 4x + 4 = 2y. by 2.
y. y= y = x2 - 2x
- 4x + 4) +2
152
Coordinate Systems and Graphing Equations
x, =
(x,,y,)
y = ax2+ bx + y=x2-2x+2
b=
= 1.
x,, x, = -b/2a x, = 4-2)/2(1) x, = 2/2 = 1 yv
+
y = x2 - 2x 2 y = (1)2 - 2(1) 2 y=1-2+2 y=-1+2 y=l
+
(1, 1).
x =2 y=x2-2x+2 y = 2 2 - 2(2) 2 y=4-4+2 y=2 (2,2).
x on y. y.
+
153
x,
Algebra
Choose x = 0 and solve for y. y = x2 - 2x 2 y = 02 - 2(0) 2 y=o-0+2 y=2 The pair is (0,2). Choose x = -1 and solve for y. y = x2 - 2x 2 y = (-1)2- 2(-1)+ 2 y=1--2+2 y=3+2 y=5 The pair is (-1,5). Choose x = 3 and solve for y. y = x2 - 2x 2 y = 32 - 2(3) + 2 y=9-6+2 y=5 The pair is (3,5).
+
+
+
+
154
Coordinate Systems and Graphing Equations
3 -5-4 -3 -2 -1
x axis
1 2 3 4 5
-3 -4
-5 -6
Y axis
9.6. Using graphing to solve quadratic equations
ax2
+ bx +
=0
0 y =ax2
y.
+ bx + by
1.
y.
155
Algebra
2. by x, =
y,. x
3.
y x
4.
x2
by y = 0). 2x2 = 2x
by
0 = 2x2 - 2x - 2 y= y. y = 2x2 - 2x - 2 by x, =
x,
=
= 2/4 = 1/2
y,.
156
+ bx +
+ 2,
Coordinate Systems and Graphing Equations
yv=2x2-2x-2 Substitute xv for x. yv = 2(1/2)2 - 2(1/2) - 2 yv = 2(1/4) - 2/2 - 2 yv=1/2-1-2 Using a common denominator of 2, the equation becomes: yv= 1/2 - 2/2 - 4/2 yv=--
1 - 2-4 2
yv = -5/2 Therefore, the vertex point is (1/2, -5/2). Choose x values near the vertex point and solve for the corresponding y values. Choose x = 1 and solve for y. y = 2x2 - 2x - 2 y = 2(1)2 - 2(1) - 2 y=2 - 2-2 y = -2
The pair is (1, -2). Choose x = -1 and solve for y. y = 2x2 - 2x - 2 y = 2(-1)2- 2(-1) - 2 y = 2 - -2 - 2 y=2 The pair is (-1,2).
157
Algebra
Choose x = 0 and solve for y. y = 2x2 - 2x - 2 y = 2(0)2 - 2(0) - 2 y=o - 0 - 2 y = -2 The pair is (0, -2). Choose x = 2 and solve for y. y = 2x2 - 2x - 2 y = 2(2)2 - 2(2) - 2 y=8 - 4 - 2 y=2 The pair is (2,2). Plot points (1/2, -5/2), (1, -2), (-1,2),(0, -2), (2,2),and sketch the parabola. 6 5 4 3 '2
41
1 n
A
158
Coordinate Systems and Graphing Equations
x by y=
x = +1.62 x = -0.62 by 0 = 2x2 - 2x - 2. x = +1.62, 0 = 2(1.62)2- 2(1.62)- 2 0=2(2.62)-3.24 - 2 0 = 5.24- 3.24- 2 0=2-2
x = -0.62, 0 = 2(-0.62)' - 2(-0.62)- 2 0 = 2(0.38) 1.24- 2 0 = 0.76 1.24- 2 0=2-2
+
+
o=o x = 1.62
0 = 2x2 - 2x - 2 x = -0.62.
159
Algebra
9.7. Using graphing to solve two linear equations with two unknown variables by by x
1.
by y. 2.
by x =0 x.
y,
y =0
by y
x
3. 4.
x
y by
5.
by
- By - 8 = 0 by by 4
160
by 2
Coordinate Systems and Graphing Equations
x+y-4=0 x - 2y - 2 = 0
x = 0, O+y-4=0 y=4
x + y - 4 = 0. y.
(0,4). y = 0,
x.
~+0-4=0 x=4 (4,O). (0,4) x = 0, - 2y - 2 = o -2y = 2
y = 0,
x - 2y - 2 = 0. y.
x.
x - 2(0) - 2 = 0 x-2=0 x=2 (2,
161
(4,O).
Algebra
-1)
(2,O). by
2
1
I Y
(3 1/3, 2/3). 2x + 2y - 8 = 0 4x - 8y - 8 = 0 x = 3 1/3 = 10/3
y = 2/3 by
10/3 + 2/3 = 4 12/3 = 4
162
Coordinate Systems and Graphing Equations
10/3- 2(2/3)= 2 -10 -3
4 =2 3
2=2
A by
A no by
do
A for by
163
Algebra
9.8. Examples of other equation forms that graph to shapes on a coordinate system
Ellipses
(0,O) r)
r)
-
A*
+-
b2
- 1
A,
An by
x y y x
164
Or
x =0 y =0
Coordinate Systems and Graphing Equations
(0,-3):
(2,0),(-2,O),(0,3) 6
5 4
-4
-5 -6
Y axis
circle
(0,O)
A=
A,
x2 + y2 =
=
A
(p, q)
by (x -
+ (y -
=
165
Algebra
x y
x =0 y =0
y x
x
(2, O), (-2,O>, (0,2)
(0, -2):
-3 -4
-5 -6
Y
166
Coordinate Systems and Graphing Equations
hyperbola
(0,O) 4 x 2 X2
+
=C
+
=
- = 1 b2
A
(p,
by
(x-pl2 b2
y
x
167
Algebra
A A
asymptotes.
+
+
=0
=0
xy=k k
k
Y
Y
L
J
X
3
168
6
25 22 76 51
20 8
8 5, 28
134 106, 127 42
39
43
50
43
105, 127 37 3 by 3
126
31
9 72 46
69, 73 76 87
13
16 22, 23
22,
168
23, 37
151
7
49
74 69
55
62
55 by
164
55
by
9 26
55 39, 76
169
98, 111 164 7 9
10, 46 10, 46 7
8 146
26, 28 138, 139
166 167 164 145
11, 139, 140
39
9 106 119 72 25
87, 92
105 61 39
127
ax2 4 bx 11 54 43
4
83 18
62
19 7 76
73
76
53 144
53 139
53
152
54 164
54
150
12, 82 9
10, 46
10, 46 150, 152, 156 139 15 12 139 on 136
164 10, 46 144 20
170
on
72 126
138 on a
90 43 137
103, 115 52
11, 49
14
6
14, 15
69 82, 150,
163
152, 155
10, 49 115, 127
92
110 96, 110, 111, 115, 119, 127 50, 62 96 unknown 96, 103, 105
83 87 90 87
43
by
160
7 134 87
120
74
26 134 87
8 13 14
145
7 6
155
12
96
6
by
17
160
171
Math series, Basic Math and Pre-Algebra Introduction
1
Chapter 1 Numbers and Their Operations
5
1.1.
6
-
1.2. 1.3. 1.4. 1.5.
1
13
1
15 22 35 43 45 46 47
1.6. 1.7. 1.8. 1.9. 1.10.
odd
1.11. 48
1.12. 51 52 53 54
1.13.
1.14. 1.15.
172
1.16. 1.17. 1.18. 1.19. 1.20.
>, <,
>, I
54 55 59 62 66
Chapter 2 Fractions
67
2.1. 2.2. 2.3,
68 69
2.4.
10
76
2.5. 2.6. 2.7.
c c
1 1
2.8 2.9
78 82 83
Chapter 3 Decimals
81
3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7.
87 90 93 95 97 98 99
Chapter 4 Percentages
101
4.1. 4.2. 4.3.
101 102
105 107 109
4.4. 4.5.
173
Chapter 5 Converting Percentages, Fractions and Decimals 5.1, 5.2. 5.3. 5.4.
5.5. 5.6.
to
111 111 112 113 113 115 117
Chapter 6 Ratios, Proportions and Variation
118
6.1. 6.2. 6.3.
118 120 121
Chapter 7 Powers and Exponents
123
7.1. 7.2. 7.3. 7.4. 7.5. 7.6. 7.7. 7.8. 7.9. 7.10. 7.11.
123 124 125 126 126 128 128 129 130 131 131
Chapter 8 Logarithms
133
8.1. 8.2. 8.3. 8.4. 8.5.
133 135 136 138 139
8.6.
140
174
Chapter 9 Roots and Radicals
145
9.1. 9.2. 9.3. 9.4. 9.5. 9.6. 9.7. 9.8. 9.9. 9.10. 9.11. 9.12.
146 147 149 152 153 154 155 155 156 156 157 158
by
Chapter 10 Important Statistical Quantities
159
10.1. 10.2. 10.3. 10.4. 10.5.
159 16 1 163 164
Index
170
Appendix
173
164
175
Table of Contents for the third book in the Master Math series, Pre-Calculus and Geometry Introduction
1
Chapter 1 Geometry
3
1.1. 1.2. 1.3. 1.4. 1.5. 1.6.
4 10 13 18 22
29 1.7.
1.8.
34 40
Chapter 2 Trigonometry
43
2.1. 2.2. 2.3.
43 44
2.4. 2.5.
52 53 of 54
2.6. 2.7.
58 60
Chapter 3 Sets and Functions
62
3.1. 3.2.
62 65
176
Chapter 4 Sequences, Progressions and Series
70
4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 4.7. 4.8. 4.9.
70 71 73 74 77 80 86 88 92
Chapter 5 Limits
94
5.1. 5.2.
94 98
Chapter 6 Introduction to the Derivative
104
6.1. 6.2. 6.3. 6.4. 6.5. 6.6. 6.7. 6.8. 6.9. 6.10.
105 110 113 114 114 118 119 120 125
p
128 131
6.11. 6.12. (minimum
134 142
6.13. 6.14. An
143
177
Chapter 7 Introduction to the Integral
146
7.1. 147 7.2. 150 152 153
7.3. 7.4. 7.5. 7.6. 7.7. 7.8. 7.9.
156 160 162 164
odd
165
by
Index
169
Appendix
175
178
