Deck 9: Quadratic Equations and Inequalities

Question 1

Solve the equation by using the square root property. Express any complex numbers using i notation.
$$x ^ { 2 } = - 25$$

A) $x = \pm 5 i$
B) $x = 5 i$
C) $x = - 12.5$
D) $x = \pm 6 \mathrm { i }$

Answer 1

A

Question 2

Solve the equation by completing the square. Express any complex numbers using i notation.
$$\frac { x ^ { 2 } } { 2 } + \frac { 1 } { 2 } x = 8$$

A) $\frac { - 1 \pm \sqrt { 65 } } { 4 }$
B) $\frac { 1 \pm \sqrt { 65 } } { 2 }$
C) $\frac { - 1 \pm \sqrt { 65 } } { 2 }$
D) $\frac { 1 \pm \sqrt { 65 } } { 4 }$

Answer 2

C

Question 3

Solve the equation by using the square root property. Express any complex numbers using i notation.
$$4 x ^ { 2 } = 60$$

A) $x = 16$
B) $x = \pm \sqrt { 15 }$
C) $x = \pm 15$
D) $x = 30$

Answer 3

B

Question 4

Solve the equation by completing the square. Express any complex numbers using i notation.
$$x ^ { 2 } + 6 x - 19 = 0$$
$\begin{array} { l l l l } \text { A) } x = 2 \sqrt { 7 } \pm 3 & \text { B) } x = - 3 \pm 2 \sqrt { 14 } & \text { C) } x = - 3 \pm 2 \sqrt { 7 } & \text { D) } x = \pm 2 \sqrt { 7 } \end{array}$

Question 5

Solve the equation by completing the square. Express any complex numbers using i notation.
$$3 x ^ { 2 } + 8 x + 1 = 0$$

A) $x = \frac { - 4 \pm \sqrt { 19 } } { 3 }$
B) $x = \frac { - 4 \pm \sqrt { 13 } } { 6 }$
C) $x = \frac { - 4 \pm \sqrt { 13 } } { 3 }$
D) $x = \frac { - 8 \pm \sqrt { 13 } } { 3 }$

Question 6

Solve the equation by using the square root property. Express any complex numbers using i notation.
$$x ^ { 2 } + 9 = 0$$

A) $x = - 4.5$
B) $x = 3 i$
C) $x = \pm 3 \mathrm { i }$
D) $x = \pm 4 \mathrm { i }$

Question 7

Solve.
The formula D = 16t2 is used to approximate the distance in feet an object falls in t seconds. A package is dropped from an airplane that is 1600 feet above the ground. How many seconds elapse before it hits the
Ground?

A)10 seconds
B)25,600 seconds
C)100 seconds
D)20 seconds

Question 8

Solve.
The time a basketball player spends in the air is called the "hang time." The vertical leap L measured in feet is related to the hang time t measured in seconds by the equation L = 4t2. Sam Bucketmaster, a spectacular
Basketball player, often displayed a leap of 3.3 feet. Find the hang time for that leap. Round your answer to the
Nearest hundredth of a second, if necessary.

A)approximately 0.825 seconds
B)approximately 0.91 seconds
C)approximately 1.65 seconds
D)approximately 43.56 seconds

Question 9

Solve.
What is the value of x if the volume of the box is 729 cubic feet?

A) $x=11$
B) $x=144$
C) $x=12$
D) $x=96$

Question 10

Solve the equation by using the square root property. Express any complex numbers using i notation.
$( x + 9 ) ^ { 2 } = 12$
B) $x = - 9 \pm 2 \sqrt { 3 }$
C) $x = - 9 \pm 2 \sqrt { 6 }$
D) $x = 2 \sqrt { 3 } \pm 9$

Question 11

Solve the equation by using the square root property. Express any complex numbers using i notation.
$( 2 x - 3 ) ^ { 2 } = 25$

A) $x = 8 , - 2$
B) $x = 1 , - 4$
C) $x = 2 , - 8$
D) $x = 4 , - 1$

Question 12

Solve the equation by completing the square. Express any complex numbers using i notation.
$$3 x ^ { 2 } + 12 x + 3 = 2$$

A) $x=\frac{-6 \pm \sqrt{39}}{3}$
B) $x = \frac { - 6 \pm \sqrt { 33 } } { 3 }$
C) $x = \frac { - 6 \pm \sqrt { 33 } } { 6 }$
D) $x = \frac { - 12 \pm \sqrt { 33 } } { 3 }$

Question 13

Solve the equation by completing the square. Express any complex numbers using i notation.
$$x ^ { 2 } + 8 x = 3$$

A) $x = - 4 \pm 2 \sqrt { 19 }$
B) $x = - 4 \pm \sqrt { 19 }$
C) $x = 4 + \sqrt { 19 }$
D) $x = - 1 \pm \sqrt { 19 }$

Question 14

Solve the equation by completing the square. Express any complex numbers using i notation.
$x^{2}+12 x=-23$

Question 15

Solve the equation by using the square root property. Express any complex numbers using i notation.
$$( 3 x + 5 ) ^ { 2 } = 7$$

A) $x = - 4 , \frac { 2 } { 3 }$
B) $x = \frac { - 5 \pm \sqrt { 7 } } { 3 }$
C) $x = \frac { 5 \pm \sqrt { 7 } } { 3 }$
D) $x = \frac { \sqrt { 7 } \pm 5 } { 3 }$

Question 16

Solve the equation by completing the square. Express any complex numbers using i notation.
$$5 x ^ { 2 } - 4 x = 9$$

A) $x = \frac { 5 } { 9 } , - 1$
B) $x = \frac { 5 } { 9 } , 1$
C) $x = \frac { 9 } { 5 } , - 1$
D) $x = \frac { 9 } { 5 } , 1$

Question 17

Solve the equation by completing the square. Express any complex numbers using i notation.
$$x ^ { 2 } + 9 = - x$$

A) $x = \frac { 1 \pm \sqrt { 35 } } { 2 }$
B) $x = \frac { - 1 \pm i \sqrt { 35 } } { 2 }$
C) $x = \frac { 1 \pm i \sqrt { 35 } } { 2 }$
D) $x = \frac { - 1 \pm \sqrt { 35 } } { 2 }$

Question 18

Solve the equation by completing the square. Express any complex numbers using i notation.
$$x ^ { 2 } + 6 x = 31$$

A) $x = - 3 \pm 2 \sqrt { 10 }$
B) $x = - 3 \pm 2 \sqrt { 20 }$
C) $x = 2 \sqrt { 10 } \pm 3$
D) $x = \pm 2 \sqrt { 10 }$

Question 19

Solve the equation by using the square root property. Express any complex numbers using i notation.
$$3 x ^ { 2 } + 7 = 2 x ^ { 2 }$$

A) $x = i \sqrt { 7 }$
B) $x = 49$
C) $x = \pm i \sqrt { 7 }$
D) $x = - 3.5$

Question 20

Solve the equation by using the square root property. Express any complex numbers using i notation.
$$x ^ { 2 } = 196$$

A) $x = 98$
B) $x = 14$
C) $x = \pm 14$
D) $x = \pm 15$

Question 21

Solve by the quadratic formula. Simplify answers. Use i notation for nonreal complex numbers.
$5 x ^ { 2 } - 4 x - 9 = 0$

A) $x = \frac { 9 } { 5 } , - 1$
B) $x = \frac { 5 } { 9 } , 1$
C) $x = \frac { 5 } { 9 } , - 1$
D) $x = \frac { 9 } { 5 } , 1$

Question 22

Use the discriminant to find what type of solutions (two rational, two irrational, one rational, or two nonreal complex)
each of the following equations has. Do not solve the equation.
$$x ^ { 2 } - 12 x + 36 = 0$$

A) one rational solution
B) two rational solutions
C) two nonreal complex solutions
D) two irrational solutions

Question 23

Use the discriminant to find what type of solutions (two rational, two irrational, one rational, or two nonreal complex)
each of the following equations has. Do not solve the equation.
$x ^ { 2 } - 2 x + 8 = 0$

A) two rational solutions
B) two irrational solutions
C) two nonreal complex solutions
D) one rational solution

Question 24

Use the discriminant to find what type of solutions (two rational, two irrational, one rational, or two nonreal complex)
each of the following equations has. Do not solve the equation.
$$7 + 2 x ^ { 2 } = 3 x$$

A) two irrational solutions
B) one rational solution
C) two nonreal complex solutions
D) two rational solutions

Question 25

Write a quadratic equation having the given solutions.
$- 6,2$

A) $x ^ { 2 } - 12 x - 4 = 0$
B) $x ^ { 2 } + 4 x - 12 = 0$
C) $x ^ { 2 } - 4 x - 12 = 0$
D) $x ^ { 2 } + 12 x - 4 = 0$

Question 26

Use the discriminant to find what type of solutions (two rational, two irrational, one rational, or two nonreal complex)
each of the following equations has. Do not solve the equation.
$$1 - 5 x ^ { 2 } = - 7 x - 3$$

A) two irrational solutions
B) two rational solutions
C) one rational solution
D) two nonreal complex solutions

Question 27

Solve by the quadratic formula. Simplify answers. Use i notation for nonreal complex numbers.
$$x ^ { 2 } = 24 x - 154$$

A) $x=\pm \sqrt{10}$
B) $x=\pm 10 \mathrm{i}$
C) $x=-12 \pm i \sqrt{10}$
D) $x=\pm \mathrm{i} \sqrt{10}$

Question 28

Use the discriminant to find what type of solutions (two rational, two irrational, one rational, or two nonreal complex)
each of the following equations has. Do not solve the equation.
$$9 x ^ { 2 } + 12 x + 4 = 0$$

A) two nonreal complex solutions
B) two rational solutions
C) two irrational solutions
D) one rational solution

Question 29

Write a quadratic equation having the given solutions.
9,10

A) $x ^ { 2 } + 19 x + 90 = 0$
B) $x ^ { 2 } - 90 x + 19 = 0$
C) $x ^ { 2 } - 19 x + 90 = 0$
D) $x ^ { 2 } + 90 x + 19 = 0$

Question 30

Solve by the quadratic formula. Simplify answers. Use i notation for nonreal complex numbers.
$$2 x ^ { 2 } + 10 x + 7 = 0$$

A) $x = \frac { - 5 \pm \sqrt { 11 } } { 4 }$
B) $x = \frac { - 5 \pm \sqrt { 39 } } { 2 }$
C) $x = \frac { - 5 \pm \sqrt { 11 } } { 2 }$
D) $x = \frac { - 10 \pm \sqrt { 11 } } { 2 }$

Question 31

Solve by the quadratic formula. Simplify answers. Use i notation for nonreal complex numbers.
$$4 x ^ { 2 } = 60$$

A) $x = \pm \sqrt { 15 }$
B) $x = \pm 15$
C) $x = 30$
D) $x = 16$

Question 32

Solve by the quadratic formula. Simplify answers. Use i notation for nonreal complex numbers.
$$x ^ { 2 } + 6 x + 34 = 0$$

A) $x= ,-8$
B) $x=-3 \pm 5 i$
C) $x=-3+5 i$
D) $x=-3 \pm 25 i$

Question 33

Use the discriminant to find what type of solutions (two rational, two irrational, one rational, or two nonreal complex)
each of the following equations has. Do not solve the equation.
$$x ^ { 2 } + 5 x + 1 = 0$$

A) two nonreal complex solutions
B) one rational solution
C) two rational solutions
D) two irrational solutions

Question 34

Use the discriminant to find what type of solutions (two rational, two irrational, one rational, or two nonreal complex)
each of the following equations has. Do not solve the equation.
$$8 x ^ { 2 } = - 5 x - 3$$

A) two nonreal complex solutions
B) one rational solution
C) two rational solutions
D) two irrational solutions

Question 35

Solve by the quadratic formula. Simplify answers. Use i notation for nonreal complex numbers.
$$3 x ^ { 2 } = - 12 x$$

A) $x = 4 , x = 0$
B) $x = 0$
C) $x = - 4 , x = 0$
D) $x = \pm 4$

Question 36

Use the discriminant to find what type of solutions (two rational, two irrational, one rational, or two nonreal complex)
each of the following equations has. Do not solve the equation.
$$x ^ { 2 } + 3 x - 4 = 0$$

A) two irrational solutions
B) two nonreal complex solutions
C) one rational solution
D) two rational solutions

Question 37

Solve by the quadratic formula. Simplify answers. Use i notation for nonreal complex numbers.
$$\begin{array} { l } x ^ { 2 } + x + 6 = 0 \\\end{array}$$

A) $x = \frac { 1 \pm \sqrt { 23 } } { 2 }$
B) $x = \frac { 1 \pm i \sqrt { 23 } } { 2 }$
C) $x = \frac { - 1 \pm \sqrt { 23 } } { 2 }$
D) $x = \frac { - 1 \pm i \sqrt { 23 } } { 2 }$

Question 38

Write a quadratic equation having the given solutions.
$- 8 , - 4$

A) $x ^ { 2 } - 32 x - 12 = 0$
B) $x ^ { 2 } - 12 x + 32 = 0$
C) $x ^ { 2 } + 32 x - 12 = 0$
D) $x ^ { 2 } + 12 x + 32 = 0$

Question 39

Solve by the quadratic formula. Simplify answers. Use i notation for nonreal complex numbers.
$$x ^ { 2 } + 7 x + 3 = 0$$

A) $x = \frac { - 7 \pm \sqrt { 37 } } { 14 }$
B) $x = \frac { 7 \pm \sqrt { 37 } } { 2 }$
C) $x = \frac { - 7 \pm \sqrt { 37 } } { 2 }$
D) $x = \frac { - 7 \pm \sqrt { 61 } } { 2 }$

Question 40

Solve by the quadratic formula. Simplify answers. Use i notation for nonreal complex numbers.
$$\frac { 1 } { x + 15 } + \frac { 1 } { x } = \frac { 1 } { 10 }$$

A) $x = \frac { - 5 \pm 25 } { 2 }$
B) $x = \frac { 5 \pm 25 } { 2 }$
C) $x = \frac { 35 \pm 25 } { 2 }$
D) $x = \frac { - 35 \pm 25 } { 2 }$

Question 41

Make an appropriate substitution in order to obtain a quadratic equation. Find all complex values for x.
$2 x - 12 x ^ { 1 / 2 } - 80 = 0$

A) $x = 10$
B) $x = 100$
C) $x = 16,100$
D) $x = 4,10$

Question 42

Solve. Express any nonreal complex numbers with i notation.
$$x ^ { 4 } - 6 x ^ { 2 } - 27 = 0$$

A) $x = \pm 3 , \pm i \sqrt { 3 }$
B) $x = 3 , i \sqrt { 3 }$
C) $x = - 9,3$
D) $x = \pm \sqrt { 3 } , \pm 3 \mathrm { i }$

Question 43

Solve. Express any nonreal complex numbers with i notation.
$$x ^ { 4 } - 22 x ^ { 2 } + 96 = 0$$

A) $x = 4 , \sqrt { 6 }$
B) $x = 16,6$
C) $x = \pm 4 , \pm \sqrt { 6 }$
D) $x = \pm 4 , \pm i \sqrt { 6 }$

Question 44

Solve for real roots.
$$x ^ { 1 / 2 } - 13 x ^ { 1 / 4 } = - 42$$

A) $x = 49,36$
B) $x = 2401,1296$
C) $x = 7,6$
D) $x = - 7 , - 6$

Question 45

Solve for real roots.
$$3 x ^ { 2 / 3 } - 8 x ^ { 1 / 3 } - 35 = 0$$

A) $x = 125$
B) $x = 125 , x = \frac { 343 } { 27 }$
C) $x = 125 , x = - \frac { 343 } { 27 }$
D) $x = - \frac { 343 } { 27 }$

Question 46

Solve.
A company that manufactures outdoor rocking chairs makes a daily profit $p$ according to the equation $p = - 100 x ^ { 2 } + 4300 x - 45,669$ , where $p$ is measured in dollars and $x$ is the number of rocking chairs made per day. Find the number of rocking chairs that must be made each day to produce a zero profit for the company. Round your answer to the nearest whole number.

A) 19 rocking chairs or 24 rocking chairs per day
B) 1914 rocking chairs or 2386 rocking chairs per day
C) 24 rocking chairs per day
D) 44 rocking chairs or 49 rocking chairs per day

Question 47

Write a quadratic equation having the given solutions.
$7 \mathbf { i } , - 7 \mathbf { i }$

A) $x ^ { 2 } + 49 = 0$
B) $x ^ { 2 } - 14 i x + 49 = 0$
C) $x ^ { 2 } - 49 = 0$
D) $( x - 7 ) ^ { 2 } = 0$

Question 48

Write a quadratic equation having the given solutions.
$\frac { 1 } { 2 } , \frac { 3 } { 4 }$

A) $8 x ^ { 2 } - 10 x + 3 = 0$
B) $8 x ^ { 2 } + 10 x + 3 = 0$
C) $8 x ^ { 2 } + 3 x + 10 = 0$
D) $8 x ^ { 2 } - 3 x + 10 = 0$

Question 49

Make an appropriate substitution in order to obtain a quadratic equation. Find all complex values for x.
$x-10 x^{1 / 2}+24=0$

A) $x = \pm \sqrt { 6 } , \pm 2$
B) $x = 6,4$
C) $x = 36,16$
D) $x = \pm 6 , \pm 4$

Question 50

Solve for real roots.
$$2 x ^ { 1 / 2 } - 11 x ^ { 1 / 4 } - 21 = 0$$

A) $x = 2401$
B) $x = 7 , - \frac { 3 } { 2 }$
C) $x = 2401 , \frac { 81 } { 16 }$
D) $x = - 7 , - 3$

Question 51

Find all valid real roots for the equation.
$$x ^ { 6 } + 56 x ^ { 3 } - 512 = 0$$

A) $x = 2 , x = - 4$
B) $x = - 2 , x = 4$
C) $x = 2 , x = 4$
D) $x = - 2 , x = - 4$

Question 52

Find all valid real roots for the equation.
$$x ^ { 6 } + 4 x ^ { 3 } - 45 = 0$$

A) $x = - \sqrt [ 3 ] { 5 } , x = - \sqrt [ 3 ] { 9 }$
B) $x = \sqrt [ 3 ] { 5 } , x = \sqrt [ 3 ] { 9 }$
C) $x = \sqrt [ 3 ] { 5 } , x = - \sqrt [ 3 ] { 9 }$
D) $x = - \sqrt [ 3 ] { 5 } , x = \sqrt [ 3 ] { 9 }$

Question 53

Solve. Express any nonreal complex numbers with i notation.
$$x ^ { 4 } - 40 x ^ { 2 } + 144 = 0$$

A) $x = 2,6$
B) $x = \pm 2 i , \pm 6 \mathrm { i }$
C) $x = \pm 2 , \pm 6$
D) $x = 4,36$

Question 54

Write a quadratic equation having the given solutions.
$$\frac { 1 } { 3 } , \frac { 1 } { 6 }$$

A) $x ^ { 2 } - \frac { 1 } { 18 } x + \frac { 1 } { 2 } = 0$
B) $x ^ { 2 } + \frac { 1 } { 2 } x + \frac { 1 } { 18 } = 0$
C) $x ^ { 2 } - \frac { 1 } { 2 } x + \frac { 1 } { 18 } = 0$
D) $x ^ { 2 } + \frac { 1 } { 18 } x + \frac { 1 } { 2 } = 0$

Question 55

Solve for real roots.
$x ^ { 2 / 3 } - 8 x ^ { 1 / 3 } + 15 = 0$

A) $x = 27,125$
B) $x = - 5 , - 3$
C) $x = 3,5$
D) $x = - 125 , - 27$

Question 56

Solve for real roots.
$$x - 4 x ^ { 1 / 2 } - 32 = 0$$

A) $x = 64$
B) $x = 128$
C) $x = 32$
D) $x = 48$

Question 57

Find all valid real roots for the equation.
$$3 x ^ { 8 } + 4 x ^ { 4 } = 32$$

A) no real roots
B) $x = \sqrt [ 4 ] { 4 }$
C) $x = \pm \frac { \sqrt [ 4 ] { 216 } } { 3 } , x = \sqrt [ 4 ] { 4 }$
D) $x = \pm \frac { \sqrt [ 4 ] { 216 } } { 3 }$

Question 58

Find all valid real roots for the equation.
$$x ^ { 8 } = 5 x ^ { 4 } - 4$$

A) $x = \pm \sqrt [ 4 ] { 1 } , x = \pm \sqrt [ 4 ] { 4 }$
B) $x = 1 , x = 4$
C) $x = \pm 1 , x = \pm \sqrt [ 4 ] { 4 }$
D) $x = \pm \sqrt [ 4 ] { 1 } , x = 4$

Question 59

Solve for real roots.
$x ^ { 2 / 5 } - x ^ { 1 / 5 } - 6 = 0$

A) $x = 243 , - 32$
B) $x = 3 , - 2$
C) $x = - 3,2$
D) $x = - 243,32$

Question 60

Solve. Express any nonreal complex numbers with i notation.
$$2 x ^ { 4 } = 65 x ^ { 2 } + 252$$

A) $x = \pm 6$
B) $x = \pm \frac { \sqrt { 14 } } { 2 } , \pm 6$
C) $x = \pm \frac { i \sqrt { 14 } } { 2 }$
D) $x = \pm \frac { \mathrm { i } \sqrt { 14 } } { 2 } , x = \pm 6$

Question 61

Make an appropriate substitution in order to obtain a quadratic equation. Find all complex values for x.
$$\left( x ^ { 2 } - 2 x \right) ^ { 2 } - 5 \left( x ^ { 2 } - 2 x \right) = 36$$

A) $x = 1 \pm \sqrt { 10 } , x = 1 \pm i \sqrt { 5 }$
B) $x = 1 \pm 2 \sqrt { 2 } , x = 1 \pm i \sqrt { 3 }$
C) $x = 1 \pm \sqrt { 10 } , x = 1 \pm i \sqrt { 3 }$
D) $x = 1 \pm 2 \sqrt { 2 } , x = 1 \pm i \sqrt { 5 }$

Question 62

Solve for the specified variable. Assume that all other variables are nonzero
$\mathrm { Ve } = \frac { 1 } { 2 } \mathrm { mv } ^ { 2 } ;$ for $\mathrm { v }$

A) $v = \pm \sqrt { 2 \mathrm { Ve } }$
B) $v = \pm \sqrt { \frac { V e } { 2 m } }$
C) $v = \frac { \pm \sqrt { 2 m V e } } { m }$
D) $v = \frac { \pm 2 \sqrt { \mathrm { Ve } } } { m }$

Question 63

Use the Pythagorean theorem to find the missing side(s).

$c = 5 , b = 4 a$ ; find $b$ and $a$

A) $\mathrm { a } = \frac { 5 \sqrt { 17 } } { 17 } , \mathrm {~b} = \frac { 20 \sqrt { 17 } } { 17 }$
B) $a = \frac { \sqrt { 85 } } { 17 } , b = \frac { \sqrt { 85 } } { 68 }$
C) $a = \frac { 5 \sqrt { 17 } } { 17 } , b = \frac { 5 \sqrt { 17 } } { 68 }$
D) $a = \frac { \sqrt { 85 } } { 17 } , b = \frac { 2 \sqrt { 85 } } { 17 }$

Question 64

Solve for the specified variable. Assume that all other variables are nonzero
$A = 3 \pi a ^ { 2 } ;$ for $a$

A) $a = \sqrt { 3 \pi \mathrm { A } }$
B) $a = \frac { \pm \sqrt { 3 \pi \mathrm { A } } } { 3 \pi }$
C) $a = \frac { \pm \sqrt { \mathrm { A } \pi } } { 3 }$
D) $a = \frac { \pm A \sqrt { 3 \pi } } { 3 \pi }$

Question 65

Solve for the specified variable. Assume that all other variables are nonzero
$$81 \left( a ^ { 2 } - b \right) + 11 = 7 F ; \text { for } a$$

A) $a = \pm \frac { \sqrt { 7 \mathrm {~F} + 81 \mathrm {~b} - 11 } } { 9 }$
B) $a = \pm \frac { \sqrt { 7 F - 11 } } { 9 } + b$
$C ) a = \pm \frac { \sqrt { 7 F + b - 11 } } { 9 }$
D) $a = \pm \frac { \sqrt { 7 F - b + 11 } } { 9 }$

Question 66

Solve for the specified variable. Assume that all other variables are nonzero
$$\mathrm { P } = \frac { 7 \mathrm { stQ } \mathrm { Q } ^ { 2 } } { 5 \mathrm { R } } ; \text { for } \mathrm { Q }$$

A) $\mathrm { Q } = \frac { 25 \mathrm { P } ^ { 2 } \mathrm { R } ^ { 2 } } { 49 \mathrm {~s} ^ { 2 } \mathrm { t } ^ { 2 } }$
B) $\mathrm { Q } = \frac { 49 \mathrm {~s} ^ { 2 } \mathrm { t } ^ { 2 } } { 25 \mathrm { P } ^ { 2 } \mathrm { R } ^ { 2 } }$
C) $\mathrm { Q } = \pm \sqrt { \frac { 7 \mathrm { Pst } } { 5 \mathrm { R } } }$
D) $\mathrm { Q } = \pm \sqrt { \frac { 5 \mathrm { PR } } { 7 \mathrm { st } } }$

Question 67

Solve.
The area of a rectangular wall in a classroom is 105 square feet. Its length is 6 feet shorter than three times its width. Find the length and width of the wall of the classroom.

A)width = 7 ft; length = 25 ft
B)width = 7 ft; length = 27 ft
C)width = 7 ft; length = 13 ft
D)width = 7 ft; length = 15 ft

Question 68

Solve for the specified variable. Assume that all other variables are nonzero
$( b - 7 ) y ^ { 2 } - 11 y + 8 z = 0$ ; for $y$

A) $y = \frac { 11 \pm \sqrt { 121 + 32 b z - 224 z } } { b - 7 }$
B) $y = \frac { 11 \pm \sqrt { 121 - 32 b z + 224 z } } { 2 b - 14 }$
C) $y = \frac { 11 \pm \sqrt { 121 - 32 b z + 224 z } } { b - 7 }$
D) $y = \frac { 11 \pm \sqrt { 121 + 32 b z - 224 z } } { 2 b - 14 }$

Question 69

Solve for the specified variable. Assume that all other variables are nonzero
$$\mathrm { L } = 2 \pi \mathrm { rh } + 2 \pi \mathrm { r } ^ { 2 } \text {; for } \mathrm { r }$$

A) $\mathrm { r } = \frac { \pi \mathrm { h } \pm \sqrt { \pi ^ { 2 } \mathrm {~h} ^ { 2 } + 2 \pi \mathrm { L } } } { 2 \pi }$
B) $r = \frac { - \pi h \pm \sqrt { \pi ^ { 2 } h ^ { 2 } - 2 \pi L } } { 2 \pi }$
C) $r = \frac { - \pi h \pm \sqrt { \pi ^ { 2 } h ^ { 2 } + 2 \pi L } } { \pi }$
D) $r = \frac { - \pi h \pm \sqrt { \pi ^ { 2 } h ^ { 2 } + 2 \pi L } } { 2 \pi }$

Question 70

Solve for the specified variable. Assume that all other variables are nonzero
$\mathrm { M } = \pi \mathrm { r } ^ { 2 } \mathrm { hd }$ ; for $\mathrm { r }$

A) $\mathrm { r } = \frac { \pm \sqrt { \mathrm { M } \pi h d } } { \pi \mathrm { hd } }$
B) $r = \frac { \pm \sqrt { \pi \mathrm { Mhd } } } { \text { hd } }$
C) $r = \frac { \pm M \sqrt { \pi h d } } { \pi h d }$
D) $r = \pm \sqrt { \pi M h d }$

Question 71

Solve for the specified variable. Assume that all other variables are nonzero
$$\mathrm { S } = \lambda \left( \mathrm { b } ^ { 2 } + \mathrm { B } ^ { 2 } \right) \mathrm { w } ; \text { for } \mathrm { b }$$

A) $b = \pm \sqrt { \frac { S - \lambda w B ^ { 2 } } { \lambda w } }$
B) $b = \pm \sqrt { \frac { S - B ^ { 2 } } { \lambda w } }$
C) $b = \pm \sqrt { \frac { S } { \lambda w } } - B ^ { 2 }$
D) $b = \pm \sqrt { \frac { S } { \lambda w } } - B$

Question 72

Solve.
A boat sailed a distance of 22 kilometers due south. It then banked to the left and sailed the second leg of the journey. Finally, it banked left again and sailed back to the starting point. The entire course was shaped like a
Right triangle with the 22-kilometer distance serving as the hypotenuse. The second leg of the journey was 5
Kilometers longer than the final leg. How long were the second and third legs of the journey? Round your
Answers to the nearest hundredth of a kilometer, if necessary.

A)The second leg was 18.26 kilometers. The third leg was 13.26 kilometers.
B)The second leg was 17.85 kilometers. The third leg was 12.85 kilometers.
C)The second leg was 20.35 kilometers. The third leg was 10.35 kilometers.
D)The second leg was 9.13 kilometers. The third leg was 6.63 kilometers.

Question 73

Solve for the specified variable. Assume that all other variables are nonzero
$x ^ { 2 } - 2 c x - 8 c ^ { 2 } = 0$ ; for $x$

A) $\mathrm { x } = 2 \mathrm { c } , - 4 \mathrm { c }$
B) $x = - 2 c , 4 c$
C) $x = \pm \sqrt { 8 c ^ { 2 } - 2 c x }$
D) $x = \pm \sqrt { 8 \mathrm { c } ^ { 2 } + 2 \mathrm { cx } }$

Question 74

Use the Pythagorean theorem to find the missing side(s).

$b = 8 , a = \sqrt { 7 }$ ; find $c$

A) $\sqrt { 15 }$
B) 57
C) $\sqrt { 57 }$
D) $\sqrt { 71 }$

Question 75

Make an appropriate substitution in order to obtain a quadratic equation. Find all complex values for x.
$$6 x ^ { - 2 } - x ^ { - 1 } - 1 = 0$$

A) $x = \frac { 1 } { 3 } , - \frac { 1 } { 2 }$
B) $x = 3 , - 2$
C) $x = - 3$ ,
D) $x = - \frac { 1 } { 3 } , -$

Question 76

Make an appropriate substitution in order to obtain a quadratic equation. Find all complex values for x.
$$\left( x ^ { 2 } - 2 x \right) ^ { 2 } - 11 \left( x ^ { 2 } - 2 x \right) + 24 = 0$$

A) $x = - 2 , - 1,8,3,4,3$
B) $x = 4,3$
C) $x = 8,3$
D) $x = - 2 , - 1,4,3$

Question 77

Solve.
A tour bus is traveling along a triangular path. The three straight lines form a right triangle. One leg of the triangle represents a distance of 144 miles. The other leg of the triangle is 96 miles shorter than the hypotenuse.
What is the length of the hypotenuse of this triangle? Of the other leg?

A)The hypotenuse is 159 miles long. The shorter leg is 62 miles long.
B)The hypotenuse is 155 miles long. The shorter leg is 58 miles long.
C)The hypotenuse is 156 miles long. The shorter leg is 60 miles long.
D)The hypotenuse is 158 miles long. The shorter leg is 59 miles long.

Question 78

Solve. Find all valid real roots for the equation.
$$2 + \frac { 5 } { 5 x - 1 } = \frac { - 2 } { ( 5 x - 1 ) ^ { 2 } }$$

A) $x = - \frac { 1 } { 5 } , x = 0$
B) $x = - \frac { 1 } { 5 } , x = \frac { 1 } { 10 }$
C) $x = - 2 , x = - \frac { 1 } { 2 }$
D) $x = - \frac { 1 } { 5 } , x = - \frac { 1 } { 10 }$

Question 79

Solve for the specified variable. Assume that all other variables are nonzero
$r m = t ^ { 2 } - m t$ ; for $t$

A) $t = \frac { m \pm \sqrt { m ^ { 2 } - 4 m r } } { 4 }$
B) $t = \frac { m \pm \sqrt { m ^ { 2 } + 4 m r } } { 2 m }$
C) $t = \frac { m \pm \sqrt { m ^ { 2 } + 4 \mathrm { rm } } } { 2 }$
D) $t = \sqrt { m r - m }$

Question 80

Solve.
The area of a triangular poster is 1480 square centimeters. Its altitude is 6 feet shorter than twice its base. Find the lengths of the altitude and base.

A)base = 40 cm; altitude = 37 cm
B)base = 20 cm; altitude = 74 cm
C)base = 40 cm; altitude = 74 cm
D)base = 80 cm; altitude = 18.5 cm