ALGEBRA 2 - cs.stcc.eduHomework List 097/099 Introductory Algebra through Applications Workbook You...
Transcript of ALGEBRA 2 - cs.stcc.eduHomework List 097/099 Introductory Algebra through Applications Workbook You...
ALGEBRA 2
Prepared by Nicole Bedinelli
2016
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Exams.
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Exams
206A 791206B Take 1Grade207A Take 2 Grade207B 792207C Take 1Grade208A Take 2 Grade208B 793209A Take 1Grade209B Take 2 Grade
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3
Course Outline
Chapter 6A: Factoring Polynomials6.1 Common Factors6.2 Factoring Trinomials with Coefficient of One6.3 Factoring Trinomials with Coefficient Other than OneQuiz 206A
Chapter 6B: Factoring Polynomials6.4 Factoring Perfect Square Trinomials and Difference of Squares6.5 Solving Quadratic Equations by FactoringQuiz 206B
Exam 1 – Test Code 791 – Chapter 6
Chapter 7A: Rational Expressions and Equations7.1 Rational Expressions7.2 Multiplication and Division of Rational ExpressionsQuiz 207A
Chapter 7B: Rational Expressions and Equations7.3 Addition and Subtraction of Rational Expressions7.4 Complex Rational ExpressionsQuiz 207B
Chapter 7C: Rational Expressions and Equations7.5 Solving Rational Expressions7.6 Ratio and ProportionQuiz 207C
Exam 2 – Test Code 792 – Chapter 7
Chapter 8A: Radical Expressions and Equations8.1 Introduction to Radical Expressions8.2 Addition and Subtraction of Radical ExpressionsQuiz 208A
Chapter 8B: Radical Expressions and Equations8.3 Multiplication and Division of Radical Expressions8.4 Solving Radical ExpressionsQuiz 208B
Chapter 9A: Quadratic Equations9.1 Solving Quadratic Equations by the Square Root Property9.3 Solving Quadratic Equations Using the Quadratic FormulaQuiz 209A
Chapter 9B: Quadratic Equations9.1 Graphing Quadratic EquationsQuiz 209B
Final Examination – Test Code 793This final examination is comprehensive, including all units covered this semester.
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Homework List 097/099
Introductory Algebra through Applications Workbook
You will find the problems under Additional Exercises
Chapter 6
6.1 p. 175 – 176 # 2, 4, 5, 8, 11, 12
6.2 p. 178 – 179 # 3, 5, 6, 9, 14, 18
6.3 p. 182 – 183 # 1, 2, 3, 4, 5, 6
Quiz 206A
6.4 p. 187 – 189 # 5, 6, 8, 13, 14, 17, 19
6.5 p. 194 – 196 # 3, 4, 5, 7, 10, 12, 14, 16, 18, 21
Quiz 206B
Exam 1 791 (Based on chapter 6)
Chapter 7
7.1 p. 199 – 201 # 1, 2, 4, 8, 10, 11, 12, 13, 14, 15, 17
7.2 p. 204 – 207 # 1, 3, 5, 8, 9, 11, 12, 14, 17
Quiz 207A
7.3 p. 211 – 213 # 1, 2, 5, 6, 14, 15, 17
7.4 p. 217 – 219 # 1, 4, 7, 8, 9, 11
Quiz 207B
7.5 p. 223 – 226 # 1, 3, 4, 5, 10, 11, 14, 16
7.6 p. 228 – 230 # 2, 5, 7, 8, 11, 12, 13
Quiz 207C
Exam 2 792 (Based on Chapter 7)
Chapter 8
8.1 p. 233 – 234 # 1 – 18
8.2 p. 236 – 238 # 1, 2, 3, 4, 6, 8, 9, 10, 13, 16
Quiz 208A
8.3 p. 241 – 244 # 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 15, 16, 18, 19, 20, 21, 22
8.4 p. 247 – 250 # 1, 2, 3, 5, 6, 8, 12, 16
Quiz 208B
Chapter 9
9.1 p. 253 – 254 # 1, 4, 6, 8, 13, 15, 17, 18
9.3 p. 262 – 264 # 1, 3, 5, 7, 9, 11, 13, 15
Quiz 209A
9.4 p. 270 – 272 # 3, 4, 6, 8, 9, 11
Quiz 209B
Exam 3 793 ( Cumulative Final)
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Chapter 6 Factoring Polynomials
6.1 Greatest Common Factor: GCF
List all the factors of the following numbers:
20 30
Which is the largest factor the two numbers have in common? Or the GCF.
Find the GCF.
Ex. 1. 40, 100 Ex. 2. 12, 16
GCF of monomials.
2x3 , and 8x5 The GCF of the coefficients 2 and 8 is 2.
Both terms have 3 factors of x in common.
So the GCF is 2x3
Find the GCF.
Ex. 1. x3, x4, x7 Ex. 2. 12x2, -16x3
6
Ex. 3. 3y6, 5y3 Ex. 4. 24x5y6, 8x4y4
To factor the GCF out of a polynomial, is to reverse the process of multiplication.
Multiply 2x(x2 + 2x – 3) Factor 2x3 + 4x2 – 6x
2x3 + 4x2 – 6x 2x( x2 + 2x – 3)
Factor the polynomial.
3x + 6 First find the GCF of the two terms 3x, and 6.
The GCF is 3. Divide each term by the GCF.� �
�+
�
�
3(x + 2)
Check by multiplying 3(x + 2) = 3x + 6
Factor Completely.
Ex. 1. 2x3 + 10x2 + 8x Ex. 2. x2 + 3x
Ex. 3. 84x2 – 56x + 28 Ex. 4. 3x4 – x2
Ex. 5. 2x – x2 + x Ex. 6. 14x3 – 7x2
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6.1 Classwork
Find the GCF
1. 8x4 , -24x2 2. –x2, -6x, -24x5
3. 8x2, -4x, 20 4. 2x2y, 4xy, 6x3y2
Factor. Check by multiplying.
5. z2- z 6. 8x4 – 24x2
7. 8x2- 4x – 20 8. 6x4 – 10x3 - 3x2
9. 8y3 – 20y2 + 12y – 16 10. x5y5 + x4y3 + x3y3 – x2y2
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6.2 Factoring Trinomials whose leading Coefficients is 1
This is reverse FOIL
(x + 2)(x + 3) = x2 + 3x + 2x + 6 = x2 + 5x + 6
x2+ 5x + 6 = ( x + 2)(x + 3) two numbers whose product is 6 and sum is 5
2 and 3
First follow the sign chart.
x2 + bx + c = (x + )(x + ) two numbers whose product is c, and sum is b
x2 – bx + c = (x - )(x - ) two numbers whose product is c and sum is b
x2 – bx – c = (x - )( x + ) two numbers whose product is c, and difference is b
* the larger absolute value number is the negative
x2 + bx – c = (x - )(x + ) two numbers whose product is c, and difference is b
* the larger absolute value number is positive
Using the chart, factor the following trinomials.
x2 + 7x + 10 so we need 2 numbers whose product is 10, and sum is 7
(x + )(x + ) using the sign chart, we know our binomials are both positive
x2 + 7x + 10 = (x + 5)(x + 2) * it does not matter which order the binomials are in
Check with FOIL (x + 5)(x + 2) = x2 + 2x + 5x + 10 = x2 + 7x + 10
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Factor the trinomials.
Ex. 1. x2 – 10x + 16 Ex. 2. x2 + 5x + 4
Ex. 3. x2 – 3x – 18 Ex. 4. y2 – 3y – 4
Ex. 5. x2 + 4x – 45 Ex. 6. t2 + t – 6
Ex. 7. x2 + 6x + 4 Ex. 8. 2x2 – 10x – 48
Ex. 9. – 9x + x2 + 14 Ex. 10. –x2 – 10x + 11
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6.2 Classwork Factoring Trinomials (a = 1)
Factor each completely.
1)
b2 + 8
b + 7 2)
n2 − 11
n + 10
3)
m2 +
m − 90 4)
n2 + 4
n − 12
5)
n2 − 10
n + 9 6)
b2 + 16
b + 64
7)
m2 + 2
m − 24 8)
x2 − 4
x + 24
9)
k2 − 13
k + 40 10)
a2 + 11
a + 18
11)
n2 −
n − 56 12)
n2 − 5
n + 6
-1-
11
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13)
b2 − 6
b + 8 14)
n2 + 6
n + 8
15)
2
n2 + 6
n − 108 16)
5
n2 + 10
n + 20
17)
2
k2 + 22
k + 60 18)
a2 −
a − 90
19)
p2 + 11
p + 10 20)
5
v2 − 30
v + 40
21)
2
p2 + 2
p − 4 22)
4
v2 − 4
v − 8
23)
x2 − 15
x + 50 24)
v2 − 7
v + 10
25)
p2 + 3
p − 18 26)
6
v2 + 66
v + 60
-2-
12
6.3 Factoring Trinomials with Leading Coefficient Other than One
Factoring ax2 + bx + c, where a ≠ 1
4x2 + 7x + 3 we see a = 4, b = 7, and c = 3
list all the factor groups of a and c, to get a sum of b
4 3 you need a combination that adds to 7, after you multiply across
4 3 3
1 1 4 3 + 4 = 7
2
2
Using the same sign chart as before, we have to set up the binomials.
4 3
1 1 we use the opposite of what is being multiplied
and the “a” factor is always the first in the parenthesis
So… (4x + 3)(x + 1) check by FOIL
(4x + 3)(x + 1) = 4x2 + 4x + 3x + 3 = 4x2 + 7x + 3
Factor.
3x2 – 5x – 2 need factors of 3 and 2, and a difference of 5
3 2
3 2
1 1
Since there are two possible choices, we use FOIL to find the correct one.
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Ex. 1. 2x2 + 7x + 3 Ex. 2. 2x2 + 23x + 56
Ex. 3. 7x2 – 11x – 6 Ex. 4. 12x2 – 4x – 1
Ex. 5. 20x2 – 3x - 2 Ex. 5. 2x2 – 3x – 5
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6.3 Classwork Factor
1. 3x2 – 5x + 2 2. 6x2 + 5x + 1
3. 5x2 – x – 18 4. 2x2 + 3x – 5
5. 4x2 + 5x + 1 6. 4x2 – 3x + 1
7. 2x2 + 3x – 5 8. 6x2 – x – 2
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6.4 Factoring Trinomial Squares and Difference of Squares
Trinomial squares x2 + 8x + 16 = (x + 4)(x + 4) = (x + 4)2
x2 – 12x + 36 = (x – 6)(x – 6) = (x – 6)2
Ex. 1. x2 – 6x + 9 Ex. 2. 2x2 – 20x + 50
Ex. 3. x2 + 18x + 81 Ex. 4. -12x + 36 + x2
Difference of squares a2 – b2 = (a – b )(a + b)
x2 – 4 = (x – 2)(x + 2) *only two terms, both perfect squares,
be a difference
4x2 – 9 = (2x + 3)(2x – 3)
Ex. 1. y2 – 36 Ex.2. 16x4 – 49
16
Ex. 3. x2 – 15 Ex. 4. 49 – 25t2
Ex. 5. 64 – x2 Ex. 6. 9w2 – 1
Ex. 7. 81x4 – 1 Ex. 8. p4 – 16
Ex. 9. x2 + 4 Ex. 10. 5 – 20x2
Factor Completely.
Ex. 1. 49c2 – d2 Ex. 2. 50x – 18x3
Ex. 3. 8x3 – 56x2 + 98x Ex. 4. 3x4 – 3
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6.4 Classwork
Factor completely. Remember to look first for a common factor.
1. x2 + 4x + 4 2. x2 + 20x + 100
3. 2x2 – 40x + 200 4. 64 – 16x + x2
5. p2 – 16 6. 4x2 – 25y2
7. 16a2 – 9 8. 24x2 – 54
9. y4 – 1 10. 5x2 - 405
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6.5 Solving Quadratic Equations by Factoring
Quadratic Equation ax2 + bx + c = 0 , a≠ 0
Principle of Zero Product (x + 3)(x – 2) = 0
x + 3 = 0 x – 2 = 0 set them equal to zero and solve
-3 -3 +2 +2
x = -3 x = 2
two solutions make this true, let’s check
Use the principle of zero product to solve the already factored equations.
Ex. 1. (x + 1)(x – 7) = 0 Ex. 2. x(4x – 7) = 0 Ex. 3. (2x – 1)(3x + 4) = 0
Use factoring to solve the quadratic equations.
x2 – 5x – 6 = 0 1. Set the equation equal to zero
(x – 6)(x+ 1) = 0 2. Factor
x – 6 = 0 x + 1 = 0 3. Set them equal to zero separate and solve
+6 +6 -1 -1 *(principle of zero product)
x = 6 x = -1
so x = 6, -1 are both solutions to the equation
19
Solve by factoring.
Ex. 1. x2 + 5x + 4 = 0 Ex. 2. x2 – 4x = 0
Ex. 3. 12x2 + 6x = 0 Ex. 4. 4x2 – 8x + 12 = 0
Ex. 5. x2 – 16 = 0 Ex. 6. 4x2 – 9 = 0
Ex. 7. x 2 – 3x = 28 Ex. 8. 9x2 = 16
Ex. 9. -2x2 + 13x – 21 = 0 Ex. 10. x(x + 7) = 18
20
6.5 Classwork
Solve using the principle of zero products.
1. (x + 9)(x – 7) = 0 2. (2x – 4)(3x + 5) = 0 3. x(x – 10) = 0
Solve by factoring and using the principle of zero products.
4. x2 + 7x – 18 = 0 5. x2 – 8x = 0
6. x2 = 16 7. 0 = 25 + x2 + 10x
8. 3x2 – 7x = 20 9. 2y2 + 12y = -10
10. 12y2 – 5y = 2 11. x2 – 5x = 18 + 2x
21
Chapter 6 Applications 090
1. A rectangle has a length 4 more then it’s width. The area is 60ft2. Find the length.
2. Four times a number added to the square of a number is equal to 5. Find such
numbers.
3. If the area of a triangle is 81ft2, and the base is twice the height. What is the base
of the triangle?
4. The hypotenuse of a right triangle is 17m. One leg is 7 more then the other leg.
What is the length of the longer leg?
22
5. The area of a triangle is 72in2. The height is 7 more than the base. What is the
height?
6. Twelve more than the square of a number is seven times the number. Find all such
numbers.
7. A rectangle has a width that is seven feet less than the length. The area is 60ft2. Find
the width.
8. A right triangle has a leg of 8in. The hypotenuse is 4 more than the other leg. What is
the length of the hypotenuse?
23
7.1 Rational Expressions
A rational expressionP
Qis an algebraic expression that can be written as a quotient of
two polynomials, P and Q, where Q ¹ 0.
Identify all numbers for which the following rational expressions are undefined.
4
x - 2undefined is when the denominator is zero
So when x – 2 = 0
x = 2 the rational expression is undefined when x = 2
Ex. 1.3x
x + 4Ex. 2.
5x
2Ex. 3.
� �
� � � � � � �
Simplify3
6=
1
2
To simplify a rational expression - Factor the numerator and denominator
- Divide out all common factors
� � � �
� � �=
� �
� � � � �=
24
Ex. 1.� � � � �
� � � �= Ex. 2.
� � � �
� � � �=
Ex. 3.� � � �
� � � �= Ex. 4.
� � �
� � � �=
Ex. 5.� � � � �
� � � � � � �= Ex. 6.
� � �
� � �=
Ex. 7.� � � � �
� � �= Ex. 8.
� � � �
� � � � � � �=
25
7.1 Classwork
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Identify the values for which the rational expression is undefined.
1) r - 27A) r = 2 B) r = -2 C) r = 0 D) None
1)
2) y + 74y - 7
A) y = 4 B) y = 7 C) y = 74 D) y = -7
2)
3) 2y - 4y2 - 64A) y = 8 B) y = 8 or y = -8 C) y = 64 D) y = 2
3)
Simplify.
4) 10xy3xy
A) 103 B) 13 C) y3y D) 10x3x
4)
5) 18m4p2
2m10p
A) 9m6p2 B) 9m6
p C) 9mp D) 9pm6
5)
6) 16x2y5
12x3y5
A) 3x4 B) 43 C) 43x D) 4x3
6)
7) -7x - 288x + 32
A) - 79 B) - 7
8C) x + 4 D) -4
7)
26
8) a2 - 8a(a + 3)(a - 8)
A) aa + 3 B) a - 8
a + 3 C) 1a + 3 D) a2
a + 3
8)
9) m2 - 25m25 - mA) -m B) m + 5 C) m D) -m - 5
9)
10) 2x + 210x2 + 18x + 8
A) 2x5x + 4 B) 1
5x + 4
C) 2x + 210x2 + 18x + 8
D) 2x + 55x + 18
10)
27
7.2 Multiplying and Dividing Rational Expressions
2
5.5
8=
2
x.x
4=
So you factor both numerators and denominators, then divide out common factors.
Ex. 1.� �
� � �∙� �
� �= Ex. 2.
� � � �
� �∙�
� �=
Ex. 3.� �
� � � � �∙� � � � �
� �= Ex. 4.
� � � � � � �
� � � �∙� � � � � � �
� � � �=
Ex. 5.� � � � � � � �
� � � �∙� � � � � �
� � �= Ex. 6.
� � � � � � �
� � � � � � � �∙� � � � � � � �
� � � � � � � �=
28
To divide rational expressions, multiply by the reciprocal, factor all numerators and
denominators. Divide out all common factors.
� �
�÷
� � �
�=
� �
�∙
�
� � �=
�
� �
Ex. 1.� � � � �
� � � � � �÷
� � � � � �
� � � � �= Ex. 2.
� � �
� �÷
� � �
� �=
Ex. 3.� � � � � � �
� � � � �÷
� � � �
� �= Ex. 4.
� � � � � � � �
� � � � � � �÷
� � � � � � � � �
� � � � � � �=
29
7.2 Classwork
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Multiply. Express the product in lowest terms.
1) 2x25
∙ 40x3
A) 16x B) 80x2
5x3C) x16 D) 16x
2
x3
1)
2) 2x2
3y2 ∙ 12y
6
x3
A) 8y4x B) 8y
3x C) xy
418 D) x
5y418
2)
3) 2p - 2p
∙ 5p2
8p - 8
A) 16p2 + 32p + 165p3
B) 10p3 - 10p2
8p2 - 8p
C) 45p D) 5p4
3)
4) k2 + 6k + 9k2 + 12k + 27
∙ k2 + 9kk2 - 2k - 15
A) k2 + 9kk - 5 B) k
k2 + 12k + 27C) kk - 5 D) 1
k - 5
4)
Divide. Express the quotient in lowest terms.
5) 2x25
÷ x315
A) x 6 B) x
537 C) 6
x D) 6x2
x3
5)
6) 4d4
k2 ÷ 2d
6
k5
A) k3
2d2B) k
2
d4C) 8d
6
k5D) 2k
3
d2
6)
30
7) 6z - y3x
÷ 4y - 24z3x - 18
A) 4x6 - x B) 6x C) 6 - x
4 D) 6 - x4x
7)
8) 7x + 74x + 4
÷ 14x + 14x + 1
A) 18 B) 14 C) 1x + 1 D) x + 1
8
8)
9) y2 - 14y + 48y2 - 64
÷ y2 - 9y2 - 5y - 24
A) (y - 6)(y - 8)(y + 8)(y - 3) B) (y + 6)(y - 8)
(y + 8)(y + 3) C) y + 6y + 3 D) y - 6
y - 3
9)
10) xx2 + 9x + 20
÷ x2 + 8xx2 + 12x + 32
A) xx + 4 B) x + 5 C) x + 8
x + 5 D) 1x + 5
10)
31
7.3 Addition and Subtraction of Rational Expressions
Like denominators.
1
5+
2
5=
3
5
Ex. 1.3
x - 1+
8
x - 1= Ex. 2.
3x
4y-
x
4y=
Ex. 3.2x 2 + 5x - 7
3x - 1+
7x 2 - 5x + 6
3x - 1= Ex. 4.
6t - 7
t - 1-
t - 2
t - 1=
The least common denominator of rational expressions.
1
2+
4
5=
5
10+
8
10=
13
10The LCD of 2 and 5 is 10.
The LCD of2
9a 2,and
1
12a 4
9a2 = 3 ×3 ×a ×a 12a 4 = 2 ×2 ×3 ×a ×a ×a ×a
The LCD is 2 ×2 ×3 ×3 ×a ×a ×a ×a = 36a 4
To find the LCD, factor each denominator completely. Then use each factor the greatest
number of times it occurs in any of the denominators.
32
Find the LCD of each group of rational expressions.
Ex. 1.1
6and
2
nEx. 2.
1
10xand
7
12x 2
Ex. 3.1
tand
3
t + 1Ex. 4.
4x + 3
3x 2 - 3xand
x - 6
x 2 - 2x + 1
Ex. 5.7p + 1
p + 2and
5p
p - 1Ex. 6.
9n + 1
2n2 + 2nand
n - 5
n2 + 2n + 1
Adding and subtracting rational expressions with unlike denominators.
3
5x 2+
x + 1
6xy= The LCD of 5x2 and 6xy is 30x2y
6y
6y×
3
5x 2+
(x + 1)
6xy×5x
5xMultiply each fraction by the missing pieces
18y
30x 2y+
5x (x + 1)
30x 2yAdd the numerators, and simplify.
18y + 5x (x + 1)
30x 2y=
18y + 5x 2 + 5x
30x 2y
33
Add or subtract the rational expressions.
Ex. 1.1
2n-
3
5n= Ex. 2.
2
7p3+
p + 3
2p=
Ex. 3.x + 2
x-
x - 4
x + 3= Ex. 4.
9
2x + 4+
x
x 2 - 4=
Ex. 5.6
x - 1-
4
1 - x= Ex. 6.
3x - 4
x - 2+
x + 1
2 - x=
Ex. 7.y
y 2 + 5Y + 6-
4Y + 1
Y 2 + 3Y + 2= Ex. 8.
7n
n2 + 4n + 3-
3n - 2
n2 + 2n + 1=
34
7.3 Classwork
Name___________________________________
Perform the indicated operation or operations. Simplify, if possible.
1) 512 x
+ 212 x
1)
2) -30m - 6
+ 5mm - 6
2)
3) m2 - 7mm - 5
+ 10m - 5
3)
The following expressions represent denominators of rational expressions. Find their LCD.4) 2(a + 7 ) and 7(a + 7 )
A) 9(a + 7 ) B) 14(a + 7 )2 C) 9(a + 7 )2 D) 14(a + 7 )4)
5) n, 5 + n, and 5 - nA) n(5 + n)(5 - n) B) 25 - n2 C) 25n2 D) n2 + 25
5)
6) r2 + 6r + 9 and r2 + 3rA) r(r + 1)(r + 3) B) r(r + 3) C) r(r + 3)2 D) (r + 3)2
6)
Perform the indicated operation. Simplify, if possible.
7) 314c
+ 910c2
A) 1214c + 10c2
B) 3(5c + 21)70c2
C) 10870c2
D) 12140c2
7)
8) 5z2
- 3z
A) 5 - 3 zz2
B) 5 z + 3z2
C) 5 + 3 zz2
D) 3 z - 5z
8)
9) 6a + 7a - 9
A) 54a - 13a(a - 9) B) 13a - 54
a(a - 9) C) 13a - 54a(9 - a) D) 54a - 13
a(9 - a)
9)
35
10) 1x - 5
+ 45 - x
A) -3x - 5 B) 5
x - 5 C) 3x - 5 D) 4
x - 5
10)
11) 26x + 36
+ 118x + 180
A) 7x + 66(6x + 6)(3x + 10) B) -5x - 54
(6x + 6)(3x + 10)
C) 5x + 5418(x + 6)(x + 10) D) 7x + 66
18(x + 6)(x + 10)
11)
12) n + 1n2 + 4n + 4
+ n + 2n2 + 11n + 18
A) 6n + 5(n + 2)2(n + 9)
B) 2n2 + 14n - 13
(n + 2)2(n + 9)
C) 2n2 + 14n + 13
(n + 9)2(n + 2)D) 2n
2 + 14n + 13(n + 2)2(n + 9)
12)
13) 55 - y
- 2y - 5
A) 105 - y B) 7
5 - y C) -35 - y D) 3
5 - y
13)
14) 12xyx2 - y2
- x - yx + y
A) x2 + 10xy + y2(x + y)(x - y) B) x
2 + 14xy + y2(x + y)(x - y)
C) -x2 + 14xy - y2(x + y)(x - y) D) -x2 + 10xy - y2
(x + y)(x + y)
14)
15) 9xx2 - 5x + 6
- 36x2 - 6x + 8
A) x - 6(x - 3)(x - 4) B) 9(x - 6)
(x - 3)(x - 4)
C) 9(x - 2)(x - 3) D) 9x - 36
(x - 2)(x - 3)(x - 4)
15)
36
7.4 Complex Rational Expressions
1
3+
3
4
7
8-
5
6
Fractions on top of fractions. Need to find the LCD.
The LCD of 3, 4, 6, and 8 is 24. Multiply the numerator and
Denominator by the LCD.
24
1×1
3+
3
4×24
1
24
1×7
8-
5
6×24
1
=8 + 18
21 - 20=
26
1= 26
1. Find LCD
2. Multiply all terms in numerator and denominator by LCD to divide out fractions.
3. Simplify
Ex. 1.
x
2+
2x
3
1
x-
x
2
Ex. 2.
1 +1
x
1 -1
x 2
37
Ex. 3.
1
5-
1
a
(5 - a)
5
Ex. 4.x 2
1
x+
2
x 2
Ex. 5.
4
x
2
x 3
Ex. 6.
1
2a2-
1
2b2
5
a+
5
b
38
7.4 Classwork
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Simplify.
1)
x9
2y7
x6
y3
A) x15
2y10B) x3
2y10C) x
3
2y4D) x
3
y4
1)
2)
4 + 2x
x4
+ 18
A) x16 B) 16 C) 16x D) 1
2)
3)
2x
+ 3y
3x
- 2y
A) x + yy - 1 B) 2y + 3x
3y - 2x C) y + xx D) -y
3)
4)
2x + 4yx
y3x2
+ 16
A) 12x B) 6x C) 12x(x2 + 2y)
(x2 + y)D) 12x
4)
5)
1a
+ 1
1a
- 1
A) 1 B) a1 - a2
C) 1 + a1 - a D) 1 - a2
5)
39
7.5 Solving Rational Equations
x
2+
x
6=
2
3choose LCD, 6
6
1×x
2+
6
1×x
6=
6
1×2
3multiply everything by LCD, divide out denominators
3x + x = 4 solve
4x
4=
4
4
x = 1
Ex. 1.4
n-
n + 1
3= 1 Ex. 2.
4
y + 2+
2
y - 1=
12
y 2 + y - 2
40
Ex. 3. x =9
x + 3+
3x
x + 3Ex. 4.
3
x-
1
x + 4=
5
x 2 + 4x
Ex. 5.4
5=
p
10Ex. 6.
2
y=
y - 4
16
Ex. 7.a
a + 3=
4
5aEx. 8.
4
w - 3=
7
w + 3
41
7.5 Classwork
Name___________________________________
Solve.
1) x5 - x9
= 2 1)
2) 4x - 2x
= 7 2)
3) 5 - xx
- 7x
= - 34 3)
4) 4x - 1
+ 42 x - 2
= 6 4)
5) 30x - 30x - 6
= 3x 5)
6) 6y + 5
- 4y - 5
= 12y2 - 25
6)
7) 91 = 36x 7)
8) 45 = 9x + 2 8)
9) x + 16
= 14x - 7 9)
10) 4x - 52x + 1
= 2x -1x + 6 10)
11) 2t = t3t - 4 11)
42
7.6 Applications
Ratios
A truck driver drives 60mi on 8 gal of gas. At the same rate, how many gallons of gas will
it take him to drive 120mi?
It would take Sue 1.5 hours to rake the leaves and it would take Bob 2hrs. How long would
it take if they worked together?
If 20 patients can be seen in 8 hours how many can be seen in 3?
43
Two brothers share a house. It would take the younger brother working alone 45 min to
clean the attic, where it would take the older brother 30 min. If the two brother worked
together, how long would it take them to clean the attic?
If it takes a painter 6 hrs to paint a room, and another worker it would take 4 hrs. How
long would it take if they worked together?
If a 16 pound turkey feeds 18 people how many people does a 4 pound turkey feed?
44
7.6 Classwork
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve.1) Dr. Wong can see 8 patients in 2 hours. At this rate, how long would it take her to see 24patients?A) 5 hr B) 96 hr C) 16 hr D) 6 hr
1)
2) Maria and Charlie can deliver 90 papers in 3 hours. How long would it take them to deliver 39papers?A) 117 hr B) 6.9 hr C) 1.7 hr D) 1.3 hr
2)
3) If a boat uses 22 gallons of gas to go 71 miles, how many miles can the boat travel on 110 gallonsof gas?A) 375 miles B) 355 miles C) 710 miles D) 14 miles
3)
4) Martha can rake the leaves in her yard in 6 hours. Her younger brother can do the job in 7 hours.How long will it take them to do the job if they work together?
A) 1342 hr B) 7 hr C) 42 hr D) 4213 hr
4)
5) Frank can type a report in 3 hours and James takes 5 hours. How long will it take the two ofthem typing together?
A) 815 hr B) 5 hr C) 158 hr D) 152 hr
5)
6) Amy can clean the house in 8 hours. When she works together with Tom, the job takes 6 hours.How long would it take Tom, working by himself, to clean the house?
A) 25 hr B) 24 12 hr C) 2 hr D) 24 hr
6)
45
8.1 Introduction to Radical Expressions
A square root of a nonnegative real number a is a number that when squared is a.
The number under the radical sign is called the radicand.
r
Evaluate the following radical expressions.
Ex. 1. √25 Ex. 2. −√16 Ex. 3. √−4
Use a calculator to approximate the value of the irrational number to the nearest
thousandth.
Ex. 1. √10 Ex. 2. √2 Ex. 3.√�
�
Squaring a Square Root Taking the Square Root of a Square
(√ � ) � = � √ � � = �
Simplify each radical expression.
Ex. 1. (√49 ) � Ex. 2. (√2 ) � Ex. 3. √4� Ex. 4. √5�
Ex. 5. √100 � � Ex. 2. � 16 � � � � Ex. 3. − � 121 � � � � �
46
Simplify radical expression.
√24 = √4 ∙ 6 = 2√6 � 12 � � � � = � 4 ∙ 3 ∙ � � ∙ � ∙ � � ∙ � = 2� � � � � 3� �
Ex. 1. √72 Ex. 2. 2√40
Ex. 3. −4√27 Ex. 4. √20 � �
Ex. 5.√� �
� � �Ex. 6. 2√25� � � �
Ex. 7. ��
�Ex. 8. �
� �
� �
Ex. 9. �� � � � �
�Ex. 10. �
� � � � � �
�
47
8.1 Classwork
Name___________________________________
Find the value of the radical expression.1) 36
A) 8 B) 18 C) 12 D) 61)
2) - 49A) -7 B) -24.5 C) -14 D) 24.5
2)
3) 5 16A) -80 B) -20 C) 80 D) 20
3)
4) -5 100A) 500 B) -50 C) -500 D) 50
4)
Use a calculator to evaluate the radical expression, rounding to the nearest thousandth.5) 10
A) -3.286 B) 3.286 C) -3.162 D) 3.1625)
6) 5 10A) 15.840 B) -15.840 C) 15.811 D) -15.811
6)
Simplify.7) ( 64 )2
A) 8 B) 32 C) 16 D) 647)
8) 52
A) 25 B) 25 C) 5 D) 10 7
8)
9) 9x2A) 9x B) 3x C) 0 D) -3x
9)
10) x10y12
A) x5y12 B) x10y6 C) x8y10 D) x5y610)
Simplify. Assume that all variables and radicands represent nonnegative values.
11) t9
A) t9 t B) t18 t C) t4 t D) t5 t11)
12) 405 x8
A) 5 x4 9 B) 405 x4 C) 9 x4 5 D) 9 x8 512)
48
13) 72 x2
A) 72 x B) 2 x2 6 C) 6 x 2 D) 6 2 x
13)
14) 24 x2yA) 2 x 6 y B) 2 xy 6 C) 2 xy2 6 D) 2 x2 6 y
14)
Simplify.
15) 149
A) 25 B) 75 C) 17 D) 35
15)
16) 964
A) 1 B) 24 C) 83 D) 38
16)
17) 15r4
A) 15r B) 15
r4C) 15r4
r4D) 15
r2
17)
18) 75x2y49
A) x 75y7 B) 5x 3y
7 C) 5 3x2y7 D) 25x 3y
18)
19) 605x2
A) 605x B) 11 5
x C) 605x2
D) 55x
19)
20) 50yz2
x4
A) 5z 2yx2
B)5 2yz2
x2C) 25z 2y
x2D) z 50y
x2
20)
49
8.2 Addition and Subtraction of Radical Expressions
Like radicals are radical expressions that have the same radicand. Unlike radicals are
radical expressions with different radicands.
Adding like terms. Adding like radicals.
4x + 5x = 9x 4√2 + 5√2 = 9√2
Add or subtract.
Ex. 1. 5√3 − 2√3 Ex. 2. 7√ � – 2 √ � − 4√ �
Ex. 3. 4√ � − 1 − √ � − 1 Ex. 4. 3√6 + 2√2
Adding and Subtracting Unlike Terms.
√12 + √27 = √4 ∙ 3 + √9 ∙ 3 = 2√3 + 3√3 = 5√3 simplify, the collect like radicals
Ex. 1. 7√50 − √72 + √32 Ex. 2. −√4� + 3√ �
Ex. 3. −3√16� + √9� Ex. 4. 8√2 − √98
Ex. 5. √4� − 4 + √9� − 9 Ex. 6. √500 − 4√800
50
8.2 Classwork
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Combine and simplify, if possible.1) 17 3 + 11 3
A) -29 3 B) -5 3 C) 6 3 D) 28 31)
2) -12 14 - 5 14A) -17 14 B) 8 14 C) -7 14 D) 16 14
2)
3) 12 10 - 8 10 - 10A) 5 10 B) -5 10 C) 3 10 D) 4 10
3)
4) -11 14x - 1 - 9 14x - 1A) -2 14x - 1 B) (-2x - 1) 14 C) (-20x - 1) 14 D) -20 14x - 1
4)
5) 4 7 + 3 175A) 7 7 B) 19 7 C) -19 7 D) 11 7
5)
6) 2 - 7 128 + 2 162A) -37 2 B) -5 292 C) -37 292 D) -5 2
6)
7) -4 125 - 9 245 - 4 20A) 510 5 B) -91 5 C) -4 5 D) -510 5
7)
8) 2 36y5 - 9y5
A) 15 y5 B) 9y2 C) 3y2 y D) 9y2 y
8)
9) 4x + 12 + x + 3A) 3x 3 B) 2 x + 3 C) 3 x + 3 D) 5x + 15
9)
10) 6 32x2 - 2 18x2 - 2x2
A) 4x 14 B) 3x 14 C) 17x 2 D) 18x 210)
51
8.3 Multiplication and Division of Radical Expressions
Multiply.
Ex. 1. √3 ∙ √5 Ex. 2. � −5√10 � � 6√2 � Ex. 3. √12 � � ∙ √3 �
Find the product. Simplify if possible. (Distribute)
Ex. 1. √8(2√3 + √2 ) Ex. 2. √� � 5 � � − 2 � Ex. 3. √6 � 3√3 − √8 �
Find the product. Simplify if Possible. (FOIL)
Ex. 1. � √� + 1 � (√� − 1) Ex. 2. � √� + 4 � � √� − 4 �
Ex. 3. (√3 + 2)� √3 − 2 � Ex. 4. (2√2 −3)� 2√2 + 3 �
52
Find the quotient and simplify.
Ex. 1.√ � �
√ �Ex. 2.
� � � � �
√ �Ex. 3.
� √ � � �
√ � � � �Ex. 4.
� � �
� � � � � �
Rationalizing the denominator. Rewrite the expression in an equivalent form that contains no
radical in its denominator.
�
√ �=
�
√ �∙√ �
√ �=
� √ �
√ � ∙√ �=
� √ �
�
Ex. 1.�
√ �Ex. 2.
�
√ �
Ex. 3. ��
� �Ex. 4. �
� �
�
Ex. 5. ��
� �Ex. 6. �
�
� �
53
8.3 Classwork
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Multiply and simplify. Assume that all variables represent nonnegative numbers.1) p ∙ p3
A) p5 B) p2 C) p4 D) p2 p
1)
2) 2x6∙ 6x2
A) 2x8 3 B) 2x4 3 C) 4x4 3 D) 2 3x8
2)
3) 7 ( 3 - 5 )A) 21 - 35 B) 8 7C) 56 D) 7 3 + 7 5
3)
4) 13( 13 + 2)A) 195 B) 13 + 26 C) 39 D) 13 + 26
4)
Multiply, then simplify the product. Assume that all variables represent positive real numbers.5) ( 7 - 7)( 2 - 4)
A) -10 14 + 28 B) 14 + 28C) 14 - 4 7 - 7 2 + 28 D) 14 - 11 2 + 28
5)
6) (5 - 3 5)2
A) 25 - 9 5 B) 70 - 30 5 C) 25 + 9 5 D) 70 + 30 56)
7) ( 7 + 1)( 7 - 1)A) 6 - 2 7 B) 8 C) 6 + 2 7 D) 6
7)
8) ( 10 + 1)( 10 - 1)A) 9 B) 9 + 2 10 C) 9 - 2 10 D) 11
8)
Find the quotient and simplify, if possible.
9) 562
A) 2 B) 562 C) 2 7 D) 112
2
9)
10) 150x4
3x
A) 10x2 2x B) 5x2 2x C) 10x 2x D) 5x 2x
10)
54
11) 272
A) 2 B) 6 2 C) 16 D) 26
11)
Rationalize the denominator. Assume that all variables represent positive real numbers.
12) 423
A) 533 B) 4 23 C) 4 2323 D) 16 23
23
12)
13) 27
A) 51 B) 2 77 C) 2 7 D) 4 7
7
13)
14) 49 2
A) 7 2 B) 49 22 C) 11 D) 7 2
2
14)
15) 98x
A) 7xx B) 7 2x
x C) 7 2x D) 7 2x
15)
55
8.4 Solving Radical Equations
1. Isolate the radical.
2. Square each side of the Equation.
3. Solve and check.
√� − 2 = 5 check: √49 − 2 = 5+2 + 2 7 − 2 = 5
√� = 7 5 = 5
(√� )� = (7)�
� = 49
Ex. 1. � � + 3 = 7 Ex. 2. √� + 1 + 3 = 0
Ex. 3. � 2 � − 1 − 12 = −7 Ex. 4. √� + 8 = 0
Ex. 5. √2 � + 1 = √5 � − 2 Ex. 6. 1 + √1 − � = �
Ex. 7. 2√� + 6 = √� � + 19 Ex. 8. √3 � + 7 + 5 = 3 �
56
Applications.
Solve the following formulas for the indicated variable.
Ex. 1. � = ��
�solve for R Ex. 2. � = �
�
� �solve for S
Ex. 3. � = � � � solve for r Ex. 4. � = √� � + � � solve for w
Solve the following problems using the Pythagorean theorem. a2+b2=c2
Ex. 1 A 14 foot ladder is leaning against a house 11 ft up. How far away is the base of the ladder
from the house on the ground?
Ex. 2. The size of a TV is given by the length of the diagonal. What is the size of a TV whose
length is 16in and width is 9in?
57
8.4 Classwork
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve and check.1) q + 4 = 5
A) 25 B) 21 C) 29 D) 811)
2) 7q - 6 = 6
A) 367 B) 6 C) 36 D) 30 7
2)
3) 4x - 10 - 7 = 0
A) 174 B) 49 C) No solution D) 594
3)
4) x + 5 = 0A) 5 B) 25 C) No solution D) -25
4)
5) 5x + 2 = 7A) 25 B) 5 C) 9 D) 1
5)
6) 7a - 2 = 5a + 7
A) 52 B) 912 C) 29 D) 92
6)
7) 2x + 15 - x = 6A) -7, -3 B) No solution C) -3 D) -7
7)
Solve the problem.
8) The period P of a pendulum in seconds is given by the formula P = 2! L19 , where L is the
length of the pendulum. Rewrite the formula by solving the equation for L.
A) L = ! 2 P B) L = 4 P2
19 !2C) L = 19 P
2
4 !2D) L = P 19
2!
8)
9) Given the area A of a circle, the formula r = A! can be used to calculate the radius of the
circle. Rewrite the formula by rationalizing the denominator on the right hand side.
A) r = !A!
B) r = ! A C) r = A!2
D) r = !AA
9)
58
Solve and check.10) The diagram below shows the side view of a plan for a slanted roof. Find the unknown length in
this roof plan.
?5 ft
8 ftA) 26 ft B) 3 10 ft C) 89 ft D) 13 ft
10)
Answer the question.11) The diagram below shows a rope connecting the top of a pole to the ground. The rope is 27 ydlong and touches the ground 13 yd from the pole.How tall is the pole? Express the answer as a simplified radical if the radicand is not a perfectsquare.
27 yd?
13 ydA) 562 yd B) 560 yd C) 4 35 yd D) 20 yd
11)
Solve the formula for the indicated letter. Assume that all variables represent nonnegative numbers.12) v2 = 2as for v
A) v = ± 2as
B) v = ±2a s C) v = ±2as D) v = ± 2as
12)
13) A = 13!r2 for r
A) r = ±3 A! B) r = ±A3! C) r = ± 3!
AD) r = ±
3A!
13)
59
9.1 & 9.3 Solving Quadratic Equations
Standard form of Quadratic Equations is: ax2 + bx + c = 0 where a≥ 1
Write the quadratic equation in standard form and identify a, b, and c.
Ex. 1. x2 + 3x – 2 = 0 Ex. 2. 2x – 3x2 = 6 Ex. 3. 4x2 = 4
Ex. 4. x2 = -5x Ex. 5. 16 = -5x2 + x Ex. 6. –x2 = 2x – 1
Solving quadratic equations. 1. Get the equation in standard form. (set equal to zero)
2. Factor to solve.
Ex. 1. x2 = 4x + 5 Ex. 2. 2x2 + 6x = -4
Ex. 3. 7x = 5x2 Ex. 4. 16 = x2
60
Solving Quadratic Equations Using the Quadratic formula.
ax2 + bx + c = 0 Quadratic formula � =� � ±√ � � � � � �
� �
2x2 + 3x – 5 = 0 � =� ( � )±� ( � )� � ( � )( � )( � � )
� ( � )
a = 2, b = 3, c = -5
� =� � ±√� � � �
�=
� � ±√� �
�=
� � ± �
�
so there are 2 solutions � =� � � �
�=
�
�= 1� =
� � � �
�=
� � �
�=
� �
�
Solve using the quadratic formula.
Ex. 1. x2 – 3x – 10 = 0 Ex. 2. x2 + 4x = 7
61
Ex. 3. x2 = x – 1 Ex. 4. 5x2 – 8x = 3
Ex. 5. x2 + 4x + 4 = 0 Ex. 6. 2x2 + 8x = -1
62
9.3 Solving Applications
Solve for c: E = mc2
Solve for t: � √ � = �
The width of a rectangle is 3 less than the length. If the area is 11 ft2, what is the length
and width?
63
An isosceles triangle has legs x cm, and a hypotenuse of 9cm. Find the length of the legs.
A triangle has a hypotenuse of 13in, and one leg is 2 in longer than the shorter leg. Find
the length of the legs.
64
9.1 Classwork
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Rewrite the quadratic equation in the form ax2 + bx + c = 0, then identify a, b, and c.1) 4x2 = 9x
A) a = 4, b = -9, c = 0 B) a = 4, b = -9C) a = 3, b = 9, c = 0 D) a = 4, b = 0, c = 9
1)
2) 8x2 + 5 = 0A) a = 8, b = 0, c = 5 B) a = 2, b = 0, c = -5C) a = 0, b = 8, c = 5 D) a = 8, b = 5, c = 0
2)
3) 6x2 = 12x - 25A) a = 6, b = 12, c = 25 B) a = 6, b = 12, c = -25C) a = 6, b = -12, c = 25 D) a = 6, b = -12, c = -25
3)
4) -9x2 + 6x = 6A) a = 9, b = 6, c = -6 B) a = -9, b = 6, c = -6C) a = 9, b = 6, c = 6 D) a = -9, b = 6, c = 6
4)
5) (5x - 8)(x) = 7A) a = 5, b = -8, c = 7 B) a = 5, b = -8, c = -7C) a = -5, b = -8, c = -7 D) a = 5, b = 8, c = 7
5)
Solve.6) x2 = 64
A) 8 B) 8, -8 C) 9, -9 D) 326)
Solve using the quadratic formula.7) x2 - x = 6
A) -2, 3 B) 1, 6 C) 2, 3 D) -2, -37)
8) x2 + 2x - 120 = 0A) 12, -10 B) -12, 1 C) 12, 10 D) -12, 10
8)
9) 15x2 + 26x + 8 = 0
A) 25, 43
B) - 215, - 12
C) 25, - 43
D) - 25, - 43
9)
65
10) 5x2 + 10x + 2 = 0
A) -5 + 355
, -5 - 355
B) -5 + 155
, -5 - 155
C) -10 + 155
, -10 - 155
D) -5 + 1510
, -5 - 1510
10)
11) 4x2 = -12x - 2
A) -3 + 72
, -3 - 72
B) -3 + 112
, -3 - 112
C) -3 + 78
, -3 - 78
D) -12 + 72
, -12 - 72
11)
12) (5x - 9)(x + 1) = 0
A) 59, -1 B) 5
9, 1 C) 9
5, -1 D) 5
9, 0
12)
13) 3x(x - 3) = 2
A) - 53
B) 9 + 1056
, 9 - 1056
C) -9 + 1056
, -9 - 1056
D) - 35
13)
14) 4 = - 12x
- 4x2
A) -12 + 52
, -12 - 52
B) -3 + 132
, -3 - 132
C) -3 + 58
, -3 - 58
D) -3 + 52
, -3 - 52
14)
Solve the problem using the quadratic formula.15) The hypotenuse of a right triangle is 15 meters long. One leg is 3 meters longer than the other.
Find the lengths of the legs.A) 12 m, 18 m B) 6 m, 9 m C) 36 m, 39 m D) 9 m, 12 m
15)
16) The length of a rectangle is 7 cm greater than the width. The area is 228 cm2. Find the length andthe width.
A) 13 cm, 20 cm B) 12 cm, 19 cm
C) 14 cm, 21 cm D) 1072
cm, 1212
cm
16)
66
9.4 Graphing Quadratic Equations in Two Variables
A quadratic equation is a parabola that opens up, or down. This is determined by a.
y = ax2 + bx + c when a is positive it opens up
y = -ax2 + bx + c when a is negative, it opens down
Determine if the following graphs open up or down.
Ex. 1. y = 6x2 – 3x – 1 Ex. 2. y = x2 + 8 Ex. 3. y = -x2 – 2x
Ex. 4. y = -5x2 + x – 1 Ex. 5. y = -x2 Ex. 6. y = 2x2 + 3x – 2
Find the vertex of the parabola.
y = 2x2 + 4x + 1 a = 2, and b = 4
1. Find the x coordinate using� �
� �
� ( � )
� ( � )= −1
2. Find the y coordinate by evaluating the equation with the x coordinate:
y = 2x2 + 4x + 1 = 2(-1)2 + 4(-1) + 1 = 2 – 4 + 1 = -1
so the vertex is (-1, -1)
Find the vertex of the parabola.
Ex. 1. y = -2x2 – 8x + 18
67
Ex. 2. y = x2 – 6x – 7
Ex. 3. y = x2 + 16
Ex. 4. y = -5x2 + 20x
Graph y = x2 + 2x + 1
68
Find the x and y intercepts of the following quadratic equations.
Ex. 1. y = x2 + 4x + 4
Ex. 2. y = 3x2 – 5x
Ex. 3. y = x2 – 16
Ex. 4. y = x2 + 4x + 5
Ex. 5. y = 2x2 – 1x – 4
69
9.4 Claswork
Name___________________________________
Find the ordered pair for the vertex.1) y = x2 - 3
A) (3, 0) B) (-3, 0) C) (0, -3) D) (0, 3)1)
2) y = 4x2 + 40x + 103A) (3, -5) B) (-3, 5) C) (-5, 3) D) (5, -3)
2)
3) y = 3x2 + 30x + 76A) (1, -5) B) (-5, 1) C) (-1, 5) D) (5, -1)
3)
4) y = 2x2 + 20x + 53A) (5, -3) B) (-3, 5) C) (3, -5) D) (-5, 3)
4)
5) y = 4x2 + 24x + 40A) (4, -3) B) (-4, 3) C) (-3, 4) D) (3, -4)
5)
Determine if the graph opens up or down, and find the y-intercepts.6) y = -x2 + 2x - 2 6)
7) y = x2 + 5 7)
8) y = -x2 - 2 8)
9) y = 4x2 - 2x 9)
10) y = -3x2 + 5x 10)
Find the x-intercepts.11) x2 - 5x = 0 11)
12) x2 + 6x - 55 = 0 12)
13) 5x2 + 10x + 2 = 0 13)
70
ANSWERS CHAPTER 6
6.1 22. 4(v + 1)(v – 2) 5. x = 0, 8
23. (x – 10)(x – 5) 6. x = -4, 4
1. 8x2 24. (v – 5)(v – 2) 7. x = -5 only
2. x 25. (p – 3)(p +6) 8. x = 5/3, 4
3. 4 26. 6(v + 10)(v + 1) 9. y = 5, -1
4. 2xy 10. y = 2/3, -1/4
5. z(z - 1) 6.3 11. x = 9, -2
6. 8x2(x2 – 3)
7. 4(2x2 – x – 5) 1. (3x – 2)(x – 1)
8. x2(6x2 – 10x – 3) 2. (2x + 1)(3x + 1)
9. 4(2y3 – 5y2 + 3y – 4) 3. (5x + 9)(x – 2)
10. x2y2(x3y3 +x2y + xy – 1) 4. (2x + 5)(x – 1)
5. (4x + 1)(x + 1)
6.2 6. Not factorable
7. (2x + 5)(x – 1)
1. (b + 7)(b + 1) 8. (2x + 1)(3x – 2)
2. (n – 10)(n – 1)
3. (m – 9)(m + 10) 6.4
4. (n – 2)(n + 6)
5. (n – 1)(n – 9) 1. (x + 2)2
6. (b + 8)2 2. (x + 10)2
7. (m + 6)(m – 4) 3. 2(x – 10)2
8. Not factorable 4. (x – 8)2
9. (k – 5)(k – 8) 5. (p + 4)(p – 4)
10. (a + 2)(a +9) 6. (2x – 5y)(2x + 5y)
11. (n + 7)(n – 8) 7. (4a – 3)(4a + 3)
12. (n – 2)(n – 3) 8. 6(2x – 3)(2x + 3)
13. (b – 4)(b – 2) 9. (y2+1)(y – 1)(y + 1)
14. (n + 2)(n +4) 10. 5(x – 9)(x + 9)
15. 2(n+ 9)(n – 6)
16. 5(n2 + 2n +4) 6.5
17. 2(k + 5)(k + 6)
18. (a – 10)(a + 9) 1. x = 9, 7
19. (p + 10)(p + 1) 2. x = 2, -5/3
20. 5(v – 2)(v – 4) 3. x = 0, 10
21. 2(p – 1 )(p + 2) 4. x = -9, 2
71
ANSWERS CHAPTER 7
7.1
1. D
2. C
3. B
4. A
5. D
6. C
7. B
8. A
9. A
10. B
7.2
1. A
2. A
3. D
4. C
5. C
6. D
7. D
8. A
9. A
10. D
7.3
1.7
12x
2. 5
3. m – 2
4. D
5. A
6. C
7. B
8. A
9. B
10. A
11. D
12. D
13. B
14. C
15. B
7.4
1. C
2. C
3. B
4. D
5. C
7.5
1.45
2
2.2
7
3. -8
4. 2
5. -54
6. 31
7. 4
8.37
4
9. -7, 13
10.29
19
11. 2, 4
7.6
1. D
2. D
3. B
4. D
5. C
6. D
72
ANSWERS CHAPTER 8&9
8.1
1. D
2. A
3. D
4. B
5. D
6. C
7. D
8. C
9. B
10. D
11. C
12. C
13. C
14. A
15. C
16. D
17. D
18. B
19. B
20. A
8.2
1. D
2. A
3. C
4. D
5. B
6. A
7. B
8. D
9. C
10. C
8.3
1. B
2. B
3. A
4. B
5. C
6. B
7. D
8. A
9. C
10. D
11. C
12. C
13. B
14. D
15. B
8.4
1. B
2. B
3. D
4. C
5. B
6. D
7. C
8. C
9. A
10. C
11. C
12. D
13. D
9.1&9.3
1. A
2. A
3. C
4. B
5. B
6. B
7. A
8. D
9. D
10. B
11. A
12. C
13. B
14. D
15. D
16. B
9.4
1. C
2. C
3. B
4. D
5. C
6. down, (0, -2)
7. up (0, 5)
8. down (0,-2)
9. up (0, 0)
10. down (0, 0)
11. (0, 0),(5, 0)
12. (-11, 0),(5, 0)
13. (-5 ± 15
5, 0)
73