Elementary and Intermediate Algebra

Marvin L. Bittinger, David J. Ellenbogen,Barbara L. Johnson

Chapter 11

Quadratic Functions and Equations

Educators


Problem 1

Complete each of the following to form a true statement.
The principle of square roots states that if $x^{2}=k$ then $x=-$_____ or $x=$_____

Melissa S.
Numerade Educator

Problem 2

Complete each of the following to form a true statement.
If $(x+5)^{2}=49,$ then $x+5=$_____ or $x+5=$_____

Melissa S.
Numerade Educator

Problem 3

Complete each of the following to form a true statement.
If $t^{2}+6 t+9=17,$ then (_____)$^{2}=17$ and
_____$=\pm \sqrt{17}$

Melissa S.
Numerade Educator

Problem 4

Complete each of the following to form a true statement.
The equations $x^{2}+8 x+$ _____ $=23$ and $x^{2}+8 x=7$ are equivalent.

Melissa S.
Numerade Educator

Problem 5

Complete each of the following to form a true statement.
The expressions $t^{2}+10 t+$_____ and $(t+$_____$)^{2}$ are equivalent.

Melissa S.
Numerade Educator

Problem 6

Complete each of the following to form a true statement.
The expressions $x^{2}-6 x+$_____ and $(x-$_____$)^{2}$ are equivalent.

Melissa S.
Numerade Educator

Problem 7

Determine the number of real-number solutions of each equation from the given graph.
$$
\begin{array}{c}
{x^{2}+x-12=0} \\
{y=x^{2}+x-12}
\end{array}
$$
(GRAPH CANNOT COPY)

Melissa S.
Numerade Educator

Problem 8

Determine the number of real-number solutions of each equation from the given graph.
$$
\begin{array}{c}
{-3 x^{2}-x-7=0} \\
{y=-3 x^{2}-x-7}
\end{array}
$$
(GRAPH CANNOT COPY)

Melissa S.
Numerade Educator

Problem 9

Determine the number of real-number solutions of each equation from the given graph.
$$
\begin{aligned}
4 x^{2}+9 =12 x \\
y =12 x-4 x^{2}-9
\end{aligned}
$$
(GRAPH CANNOT COPY)

Melissa S.
Numerade Educator

Problem 10

Determine the number of real-number solutions of each equation from the given graph.
$$
\begin{aligned}
2 x^{2}+3 =6 x \\
y =2 x^{2}+3-6 x
\end{aligned}
$$
(GRAPH CANNOT COPY)

Melissa S.
Numerade Educator

Problem 11

Determine the number of real-number solutions of each equation from the given graph.
$$
\begin{aligned}
&f(x)=0\\
&y=f(x)
\end{aligned}
$$
(GRAPH CANNOT COPY)

Melissa S.
Numerade Educator

Problem 12

Determine the number of real-number solutions of each equation from the given graph.
$$
\begin{aligned}
&f(x)=0\\
&y=f(x)
\end{aligned}
$$
(GRAPH CANNOT COPY)

Melissa S.
Numerade Educator

Problem 13

Solve.
$$
x^{2}=100
$$

Melissa S.
Numerade Educator

Problem 14

Solve.
$$
t^{2}=144
$$

Melissa S.
Numerade Educator

Problem 15

Solve.
$$
p^{2}-50=0
$$

Melissa S.
Numerade Educator

Problem 16

Solve.
$$
c^{2}-8=0
$$

Melissa S.
Numerade Educator

Problem 17

Solve.
$$
4 x^{2}=20
$$

Melissa S.
Numerade Educator

Problem 18

Solve.
$$
7 x^{2}=21
$$

Melissa S.
Numerade Educator

Problem 19

Solve.
$$
x^{2}=-4
$$

Melissa S.
Numerade Educator

Problem 20

Solve.
$$
x^{2}=-9
$$

Melissa S.
Numerade Educator

Problem 21

Solve.
$$
9 x^{2}-16=0
$$

Melissa S.
Numerade Educator

Problem 22

Solve.
$$
25 x^{2}-4=0
$$

Melissa S.
Numerade Educator

Problem 23

Solve.
$$
5 t^{2}-3=4
$$

Melissa S.
Numerade Educator

Problem 24

Solve.
$$
3 t^{2}-1=6
$$

Melissa S.
Numerade Educator

Problem 25

Solve.
$$
4 d^{2}+81=0
$$

Melissa S.
Numerade Educator

Problem 26

Solve.
$$
25 y^{2}+16=0
$$

Melissa S.
Numerade Educator

Problem 27

Solve.
$$
(x-1)^{2}=49
$$

Melissa S.
Numerade Educator

Problem 28

Solve.
$$
(x+2)^{2}=25
$$

Melissa S.
Numerade Educator

Problem 29

Solve.
$$
(a-13)^{2}=18
$$

Melissa S.
Numerade Educator

Problem 30

Solve.
$$
(a+5)^{2}=8
$$

Melissa S.
Numerade Educator

Problem 31

Solve.
$$
(x+1)^{2}=-9
$$

Melissa S.
Numerade Educator

Problem 32

Solve.
$$
(x-1)^{2}=-49
$$

Melissa S.
Numerade Educator

Problem 33

Solve.
$$
\left(y+\frac{3}{4}\right)^{2}=\frac{17}{16}
$$

Melissa S.
Numerade Educator

Problem 34

Solve.
$$
\left(t+\frac{3}{2}\right)^{2}=\frac{7}{2}
$$

Melissa S.
Numerade Educator

Problem 35

Solve.
$$
x^{2}-10 x+25=64
$$

Melissa S.
Numerade Educator

Problem 36

Solve.
$$
x^{2}-6 x+9=100
$$

Melissa S.
Numerade Educator

Problem 37

Let $f(x)=x^{2} .$ Find $x$ such that $f(x)=19$.

Melissa S.
Numerade Educator

Problem 38

Let $f(x)=x^{2} .$ Find $x$ such that $f(x)=11$.

Melissa S.
Numerade Educator

Problem 39

Let $f(x)=(x-5)^{2} .$ Find $x$ such that $f(x)=16$.

Melissa S.
Numerade Educator

Problem 40

Let $g(x)=(x-2)^{2} .$ Find $x$ such that $g(x)=25$.

Melissa S.
Numerade Educator

Problem 41

Let $F(t)=(t+4)^{2} .$ Find $t$ such that $F(t)=13$

Melissa S.
Numerade Educator

Problem 42

Let $f(t)=(t+6)^{2} .$ Find $t$ such that $f(t)=15$.

Melissa S.
Numerade Educator

Problem 43

Let $g(x)=x^{2}+14 x+49 .$ Find $x$ such that $g(x)=49$.

Melissa S.
Numerade Educator

Problem 44

Let $F(x)=x^{2}+8 x+16 .$ Find $x$ such that $F(x)=9$.

Melissa S.
Numerade Educator

Problem 45

Replace the blanks in each equation with constants to complete the square and form a true equation.
$x^{2}+16 x+$_____$=(x+$_____$)^{2}$

Melissa S.
Numerade Educator

Problem 46

Replace the blanks in each equation with constants to complete the square and form a true equation.
$x^{2}+8 x+$_____$=(x+$_____$)^{2}$

Melissa S.
Numerade Educator

Problem 47

Replace the blanks in each equation with constants to complete the square and form a true equation.
$t^{2}-10 t+$_____$=(t-$_____$)^{2}$

Melissa S.
Numerade Educator

Problem 48

Replace the blanks in each equation with constants to complete the square and form a true equation.
$t^{2}-6 t+$_____$=(t-$_____$)^{2}$

Melissa S.
Numerade Educator

Problem 49

Replace the blanks in each equation with constants to complete the square and form a true equation.
$t^{2}-2 t+$_____$=(t-$_____$)^{2}$

Melissa S.
Numerade Educator

Problem 50

Replace the blanks in each equation with constants to complete the square and form a true equation.
$x^{2}+2 x+$_____$=(x+$_____$)^{2}$

Melissa S.
Numerade Educator

Problem 51

Replace the blanks in each equation with constants to complete the square and form a true equation.
$x^{2}+3 x+$_____$=(x+$_____$)^{2}$

Melissa S.
Numerade Educator

Problem 52

Replace the blanks in each equation with constants to complete the square and form a true equation.
$t^{2}-9 t+$_____$=(t-$_____$)^{2}$

Melissa S.
Numerade Educator

Problem 53

Replace the blanks in each equation with constants to complete the square and form a true equation.
$x^{2}+\frac{2}{5} x+$_____$=(x+$_____$)^{2}$

Melissa S.
Numerade Educator

Problem 54

Replace the blanks in each equation with constants to complete the square and form a true equation.
$x^{2}+\frac{2}{3} x+$_____$=(x+$_____$)^{2}$

Melissa S.
Numerade Educator

Problem 55

Replace the blanks in each equation with constants to complete the square and form a true equation.
$t^{2}-\frac{5}{6} t+$_____$=(t-$_____$)^{2}$

Melissa S.
Numerade Educator

Problem 56

Replace the blanks in each equation with constants to complete the square and form a true equation.
$t^{2}-\frac{5}{3} t+$_____$=(t-$_____$)^{2}$

Melissa S.
Numerade Educator

Problem 57

Solve by completing the square. Show your work.
$$
x^{2}+6 x=7
$$

Melissa S.
Numerade Educator

Problem 58

Solve by completing the square. Show your work.
$$
x^{2}+8 x=9
$$

Melissa S.
Numerade Educator

Problem 59

Solve by completing the square. Show your work.
$$
t^{2}-10 t=-23
$$

Melissa S.
Numerade Educator

Problem 60

Solve by completing the square. Show your work.
$$
t^{2}-4 t=-1
$$

Melissa S.
Numerade Educator

Problem 61

Solve by completing the square. Show your work.
$$
x^{2}+12 x+32=0
$$

Melissa S.
Numerade Educator

Problem 62

Solve by completing the square. Show your work.
$$
x^{2}+16 x+15=0
$$

Melissa S.
Numerade Educator

Problem 63

Solve by completing the square. Show your work.
$$
t^{2}+8 t-3=0
$$

Melissa S.
Numerade Educator

Problem 64

Solve by completing the square. Show your work.
$$
t^{2}+6 t-5=0
$$

Melissa S.
Numerade Educator

Problem 65

Complete the square to find the $x$ -intercepts of each function given by the equation listed.
$$
f(x)=x^{2}+6 x+7
$$

Melissa S.
Numerade Educator

Problem 66

Complete the square to find the $x$ -intercepts of each function given by the equation listed.
$$
f(x)=x^{2}+10 x-2
$$

Melissa S.
Numerade Educator

Problem 67

Complete the square to find the $x$ -intercepts of each function given by the equation listed.
$$
g(x)=x^{2}+9 x-25
$$

Melissa S.
Numerade Educator

Problem 68

Complete the square to find the $x$ -intercepts of each function given by the equation listed.
$$
g(x)=x^{2}+5 x+2
$$

Melissa S.
Numerade Educator

Problem 69

Complete the square to find the $x$ -intercepts of each function given by the equation listed.
$$
f(x)=x^{2}-10 x-22
$$

Melissa S.
Numerade Educator

Problem 70

Complete the square to find the $x$ -intercepts of each function given by the equation listed.
$$
f(x)=x^{2}-8 x-10
$$

Melissa S.
Numerade Educator

Problem 71

Solve by completing the square. Remember to first divide, as in Example $11,$ to make sure that the coefficient of $x^{2}$ is 1.
$$
9 x^{2}+18 x=-8
$$

Melissa S.
Numerade Educator

Problem 72

Solve by completing the square. Remember to first divide, as in Example $11,$ to make sure that the coefficient of $x^{2}$ is 1.
$$
4 x^{2}+8 x=-3
$$

Melissa S.
Numerade Educator

Problem 73

Solve by completing the square. Remember to first divide, as in Example $11,$ to make sure that the coefficient of $x^{2}$ is 1.
$$
3 x^{2}-5 x-2=0
$$

Melissa S.
Numerade Educator

Problem 74

Solve by completing the square. Remember to first divide, as in Example $11,$ to make sure that the coefficient of $x^{2}$ is 1.
$$
2 x^{2}-5 x-3=0
$$

Melissa S.
Numerade Educator

Problem 75

Solve by completing the square. Remember to first divide, as in Example $11,$ to make sure that the coefficient of $x^{2}$ is 1.
$$
5 x^{2}+4 x-3=0
$$

Melissa S.
Numerade Educator

Problem 76

Solve by completing the square. Remember to first divide, as in Example $11,$ to make sure that the coefficient of $x^{2}$ is 1.
$$
4 x^{2}+3 x-5=0
$$

Melissa S.
Numerade Educator

Problem 77

Find the $x$ -intercepts of the function given by $f(x)=4 x^{2}+2 x-3$.

Melissa S.
Numerade Educator

Problem 78

Find the $x$ -intercepts of the function given by $f(x)=3 x^{2}+x-5$.

Melissa S.
Numerade Educator

Problem 79

Find the $x$ -intercepts of the function given by $g(x)=2 x^{2}-3 x-1$.

Melissa S.
Numerade Educator

Problem 80

Find the $x$ -intercepts of the function given by $g(x)=3 x^{2}-5 x-1$.

Melissa S.
Numerade Educator

Problem 81

Use $A=P(1+r)^{t}$ to find the interest rate. Refer to Example $12 .$
2000 dollars grows to 2420 dollars in 2 years

Melissa S.
Numerade Educator

Problem 82

Use $A=P(1+r)^{t}$ to find the interest rate. Refer to Example $12 .$
1000 dollars grows to 1440 dollars in 2 years

Melissa S.
Numerade Educator

Problem 83

Use $A=P(1+r)^{t}$ to find the interest rate. Refer to Example $12 .$
6250 dollars grows to 6760 dollars in 2 years

Melissa S.
Numerade Educator

Problem 84

Use $A=P(1+r)^{t}$ to find the interest rate. Refer to Example $12 .$
6250 dollars grows to 7290 dollars in 2 years

Melissa S.
Numerade Educator

Problem 85

Use $s=16 t^{2}$. Refer to Example 13 and neglect air resistance.
The Grand Canyon skywalk is 4000 ft above the Colorado River. How long will it take a stone to fall from the skywalk to the river? Source: www.grandcanyonskywalk.com (IMAGE CANNOT COPY)

Melissa S.
Numerade Educator

Problem 86

Use $s=16 t^{2}$. Refer to Example 13 and neglect air resistance.
The Sears Tower in Chicago is 1454 ft tall. How long would it take an object to fall freely from the top?

Melissa S.
Numerade Educator

Problem 87

Use $s=16 t^{2}$. Refer to Example 13 and neglect air resistance.
At $2063 \mathrm{ft}$, the KVLY-TV tower in North Dakota is the tallest supported tower in the United States. How long would it take an object to fall freely from the top? Source: North Dakota Tourism Division.

Melissa S.
Numerade Educator

Problem 88

Use $s=16 t^{2}$. Refer to Example 13 and neglect air resistance.
El Capitan in Yosemite National Park is 3593 ft high. How long would it take a carabiner to fall freely from the top? Source: Guinness World Records 2008 (IMAGE CANNOT COPY)

Melissa S.
Numerade Educator

Problem 89

Explain in your own words a sequence of steps that can be used to solve any quadratic equation in the quickest way.

Melissa S.
Numerade Educator

Problem 90

Describe how to write a quadratic equation that can be solved algebraically but not graphically.

Melissa S.
Numerade Educator

Problem 91

To prepare for Section $11.2,$ review evaluating expressions and simplifying radical expressions (Sections $1.8,10.3$ and $10.8)$ Evaluate. [ 1.8]
$$
b^{2}-4 a c, \text { for } a=3, b=2, \text { and } c=-5
$$

Melissa S.
Numerade Educator

Problem 92

To prepare for Section $11.2,$ review evaluating expressions and simplifying radical expressions (Sections $1.8,10.3$ and $10.8)$ Evaluate. [ 1.8]
$$
b^{2}-4 a c, \text { for } a=1, b=-1, \text { and } c=4
$$

Melissa S.
Numerade Educator

Problem 93

To prepare for Section $11.2,$ review evaluating expressions and simplifying radical expressions (Sections $1.8,10.3$ and $10.8)$ Simplify. $[10.3],[10.8]$
$$
\sqrt{200}
$$

Melissa S.
Numerade Educator

Problem 94

To prepare for Section $11.2,$ review evaluating expressions and simplifying radical expressions (Sections $1.8,10.3$ and $10.8)$ Simplify. $[10.3],[10.8]$
$$
\sqrt{96}
$$

Melissa S.
Numerade Educator

Problem 95

To prepare for Section $11.2,$ review evaluating expressions and simplifying radical expressions (Sections $1.8,10.3$ and $10.8)$ Simplify. $[10.3],[10.8]$
$$
\sqrt{-4}
$$

Melissa S.
Numerade Educator

Problem 96

To prepare for Section $11.2,$ review evaluating expressions and simplifying radical expressions (Sections $1.8,10.3$ and $10.8)$ Simplify. $[10.3],[10.8]$
$$
\sqrt{-25}
$$

Melissa S.
Numerade Educator

Problem 97

To prepare for Section $11.2,$ review evaluating expressions and simplifying radical expressions (Sections $1.8,10.3$ and $10.8)$ Simplify. $[10.3],[10.8]$
$$
\sqrt{-8}
$$

Melissa S.
Numerade Educator

Problem 98

To prepare for Section $11.2,$ review evaluating expressions and simplifying radical expressions (Sections $1.8,10.3$ and $10.8)$ Simplify. $[10.3],[10.8]$
$$
\sqrt{-24}
$$

Melissa S.
Numerade Educator

Problem 99

What would be better: to receive $3 \%$ interest every 6 months or to receive $6 \%$ interest every 12 months? Why?

Melissa S.
Numerade Educator

Problem 100

Example 12 was solved with a graphing calculator by graphing each side of
$$
4410=4000(1+r)^{2}
$$
How could you determine, from a reading of the problem, a suitable viewing window?

Melissa S.
Numerade Educator

Problem 101

Find $b$ such that each trinomial is a square.
$$
x^{2}+b x+81
$$

Melissa S.
Numerade Educator

Problem 102

Find $b$ such that each trinomial is a square.
$$
x^{2}+b x+49
$$

Melissa S.
Numerade Educator

Problem 103

If $f(x)=2 x^{5}-9 x^{4}-66 x^{3}+45 x^{2}+280 x$ and $x^{2}-5$ is a factor of $f(x),$ find all $a$ for which $f(a)=0$.

Melissa S.
Numerade Educator

Problem 104

If $f(x)=\left(x-\frac{1}{3}\right)\left(x^{2}+6\right)$ and $g(x)=$ $\left(x-\frac{1}{3}\right)\left(x^{2}-\frac{2}{3}\right),$ find all $a$ for which $(f+g)(a)=0$.

Melissa S.
Numerade Educator

Problem 105

A barge and a fishing boat leave a dock at the same time, traveling at a right angle to each other. The barge travels $7 \mathrm{km} / \mathrm{h}$ slower than the fishing boat. After 4 hr, the boats are $68 \mathrm{km}$ apart. Find the speed of each boat. (IMAGE CANNOT COPY)

Melissa S.
Numerade Educator

Problem 106

Find three consecutive integers such that the square of the first plus the product of the other two is 67 .

Melissa S.
Numerade Educator