Elementary and Intermediate Algebra

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Problem 13

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

Melissa S.

Problem 14

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

Melissa S.

Problem 15

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

Melissa S.

Problem 16

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

Melissa S.

Problem 17

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

Melissa S.

Problem 18

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

Melissa S.

Problem 19

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

Melissa S.

Problem 20

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

Melissa S.

Problem 21

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

Melissa S.

Problem 22

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

Melissa S.

Problem 23

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

Melissa S.

Problem 24

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

Melissa S.

Problem 25

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

Melissa S.

Problem 26

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

Melissa S.

Problem 27

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

Melissa S.

Problem 28

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

Melissa S.

Problem 29

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

Melissa S.

Problem 30

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

Melissa S.

Problem 31

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

Melissa S.

Problem 32

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

Melissa S.

Problem 33

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

Melissa S.

Problem 34

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

Melissa S.

Problem 35

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

Melissa S.

Problem 36

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

Melissa S.

Problem 37

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

Melissa S.

Problem 38

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

Melissa S.

Problem 39

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

Melissa S.

Problem 40

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

Melissa S.

Problem 41

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

Melissa S.

Problem 42

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

Melissa S.

Problem 43

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

Melissa S.

Problem 44

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

Melissa S.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Problem 57

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

Melissa S.

Problem 58

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

Melissa S.

Problem 59

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

Melissa S.

Problem 60

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

Melissa S.

Problem 61

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

Melissa S.

Problem 62

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

Melissa S.

Problem 63

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

Melissa S.

Problem 64

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

Melissa S.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Problem 77

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

Melissa S.

Problem 78

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

Melissa S.

Problem 79

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

Melissa S.

Problem 80

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

Melissa S.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Problem 90

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

Melissa S.

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.

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.

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.

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.

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.

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.

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.

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.

Problem 99

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

Melissa S.

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.

Problem 101

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

Melissa S.

Problem 102

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

Melissa S.

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.

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.

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.