Deck 6: Quadratic Equations and Inequalities

Question 1

Write $\sqrt { 5 } \sqrt { - 49 }$ in terms of i , perform the indicated operations, and simplify.

A) $7 i \sqrt { 5 }$
B) $7 \sqrt { 5 }$
C) $- 7 i \sqrt { 5 }$
D) $7 \sqrt { 5 i }$
E) $- 7 \sqrt { 5 }$

Answer 1

$7 i \sqrt { 5 }$

Question 2

$| a + b i | =$ __________

A) $a + b$
B) $\sqrt { a + b }$
C) $\sqrt { a ^ { 2 } - b ^ { 2 } }$
D) $\sqrt { a - b }$
E) $\sqrt { a ^ { 2 } + b ^ { 2 } }$

Answer 2

$\sqrt { a ^ { 2 } + b ^ { 2 } }$

Question 3

Write $- 6 \sqrt { - 48 }$ in terms of i and simplify.

A) $- 24 i \sqrt { 3 }$
B) $24 i \sqrt { 3 i }$
C) $- 24 \sqrt { 3 }$
D) $- 6 i \sqrt { 3 }$
E) $24 i \sqrt { 3 }$

Answer 3

$- 24 i \sqrt { 3 }$

Question 4

Factor the trinomial square and use the square root method to solve the equation. $3 x ^ { 2 } + 7 x + 1 = 2$

A) $\frac { ( - 10 + \sqrt { 61 } ) } { 7 }$ , $\frac { ( - 10 - \sqrt { 61 } ) } { 7 }$
B) $\frac { ( - 6 + \sqrt { 61 } ) } { 7 }$ , $\frac { ( - 6 - \sqrt { 61 } ) } { 7 }$
C) $\frac { ( - 7 + \sqrt { 61 } ) } { 6 }$ , $\frac { ( - 7 - \sqrt { 61 } ) } { 6 }$
D) $\frac { ( - 8 \sqrt { 61 } ) } { 7 }$ , $\frac { ( - 8 - \sqrt { 61 } ) } { 7 }$

Question 5

The complex conjugate of is _________.

Question 6

Write $\sqrt { - 49 } \sqrt { - 9 }$ in terms of i , perform the indicated operations, and simplify.

A)28
B)-28
C)-28 i
D)28 + i
E)28 i

Question 7

Write the expression in $a + b i$ form. $\frac { \sqrt { 3 } + \sqrt { 2 } i } { \sqrt { 3 } - \sqrt { 2 } i }$

A) $\frac { 6 } { 5 } + \frac { 2 } { 5 } i$
B) $\frac { 1 } { 5 } - 2 \frac { \sqrt { 6 } } { 5 } i$
C) $\frac { 5 } { 6 } + \frac { \sqrt { 6 } } { 5 } i$
D) $\frac { 1 } { 5 } + 2 \frac { \sqrt { 6 } } { 5 } i$
E) $\frac { 5 } { 6 } - \frac { \sqrt { 6 } } { 5 } i$

Question 8

Write the expression in a + bi form. $\frac { 2 i } { 6 + i }$

A) $\frac { 2 } { 37 } + \frac { 12 } { 37 } i$
B) $\frac { 37 } { 2 } - \frac { 37 } { 12 } i$
C) $\frac { 12 } { 37 } - \frac { 2 } { 37 } i$
D) $\frac { 37 } { 12 } - \frac { 37 } { 2 } i$

Question 9

Simplify the expression. Enter the answer in a + bi form. $( 6 + \sqrt { - 272 } ) + ( - 2 - \sqrt { - 17 } )$

A) $4 + 3 \sqrt { 17 i }$
B) $$4 - 3 \sqrt { 17 i }$$
C) $( 4 - 3 \sqrt { 17 } ) i$
D) $( 4 + 3 \sqrt { 17 } ) i$

Question 10

Simplify the expression. Enter the answer in a + bi form. $( 5 + 8 i ) + ( 1 - 3 i )$

A) $6 - 5 i$
B) $6 + 5 i$
C) $29 + 7 i$
D) $29 - 7 i$

Question 11

Perform the operation. Write the answer in $a + b i$ form. $( 9 - 4 i ) - ( 3 + 6 i )$

A) $6 + 10 i$
B) $6 - 10 i$
C) $6 i - 10$
D) $6 i + 10$

Question 12

Use the factoring method to solve the equation. $6 x ^ { 2 } + 11 x + 3 = 0$

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

Question 13

Simplify the expression. Enter the answer in a + bi form. $( 7 + \sqrt { - 15 } ) ( 3 - \sqrt { - 17 } )$

A) $21 - \sqrt { 255 } + ( 3 \sqrt { 15 } + 7 \sqrt { 17 } ) i$
B) $21 + \sqrt { 255 } + ( 3 \sqrt { 15 } + 7 \sqrt { 17 } ) i$
C) $21 + \sqrt { 255 } + ( \sqrt [ 3 ] { 15 } - 7 \sqrt { 17 } ) i$
D) $21 - \sqrt { 255 } + ( \sqrt [ 3 ] { 15 } - 7 \sqrt { 17 } ) i$

Question 14

Write $\frac { \sqrt { - 96 } } { \sqrt { - 6 } }$ in terms of i , perform the indicated operations, and simplify.

A)8 i
B)-8 i
C)-8
D)8
E)8 + i

Question 15

Simplify the expression. Enter the answer in a + bi form. $( - 5 + \sqrt { - 15 } ) ( 1 - \sqrt { - 15 } )$

A) $10 - 6 \sqrt { 15 i }$
B) $10 + 6 \sqrt { 15 i }$
C) $- 20 + 6 \sqrt { 15 i }$
D) $- 20 - 6 \sqrt { 15 } i$

Question 16

Write $\sqrt { - 4 }$ in terms of i and simplify.

A)7
B)7+ i
C)7 i
D)-7 i
E)-7

Question 17

Simplify the expression. Enter the answer in a + bi form. $( 5 + i ) ( 2 - 3 i )$

A) $7 + 13 i$
B) $13 - 13 i$
C) $13 + 13 i$
D) $7 - 13 i$

Question 18

Write $\sqrt { - \frac { 25 } { 4 } }$ in terms of i and simplify.

A) $\frac { 5 } { 2 }$
B) $- \frac { 5 } { 2 } + i$
C) $\frac { 5 } { 2 } + i$
D) $- \frac { 5 i } { 2 }$
E) $\frac { 5 i } { 2 }$

Question 19

Write the expression in a + bi form. $\frac { 6 - 5 i } { 6 + 5 i }$

A) $\frac { 11 } { 61 } - \frac { 60 } { 61 } i$
B) $\frac { 11 } { 61 } - \frac { 61 } { 60 } i$
C) $\frac { 61 } { 11 } - \frac { 60 } { 61 } i$
D) $\frac { 60 } { 61 } - \frac { 11 } { 61 } i$

Question 20

$a + b i$ and $a - b i$ are called complex __________.

A)part
B)conjugates
C)line

Question 21

Factor the trinomial square and use the square root method to solve the equation. $x ^ { 2 } - 7 x + 5 = 5$

A) $0$
B) $\pm 7$
C) $0 , - 7$
D) $0,7$
E) $7$

Question 22

Use the quadratic formula to find all real solutions of the equation. $x ^ { 2 } - 11 x + 24 = 0$

A)x = 6, x = 16
B)x = 3, x = -8
C)x = 3, x = 16
D)x = -3, x = 8
E)x = 3, x = 8

Question 23

Solve the equation by completing the square. $x ^ { 2 } - 4 x - 5 = 0$

A) $x = - 2$ , $x = 5$
B) $x = 4$ , $x = 5$
C) $x = 1$ , $x = 5$
D) $x = - 1$ , $x = 5$
E) $x = - 10$ , $x = 2$

Question 24

Solve the equation. $10 ( m + 3 ) ^ { - 2 } - 29 ( m + 3 ) ^ { - 1 } + 9 = 0$

A) $\frac { 29 \pm \sqrt { 481 } } { 18 } - 3$
B) $\frac { 29 \pm \sqrt { 481 } } { 18 } + 3$
C) $$\frac { 29 \pm \sqrt { 481 } } { 20 } + 3$$
D) $\frac { 29 \pm \sqrt { 481 } } { 20 } - 3$

Question 25

Solve the equation. $z - 5 + \frac { 6 } { z } = 0$

A) $$z = - 2 , - 3$$
B) $z = 1 , - 5$
C) $z = - 2,3$
D) $z = 2,3$
E) $z = 2 , - 3$

Question 26

Solve the equation by completing the square. $x ^ { 2 } = 32 - 4 x$

A) $x = 5$ , $x = 10$
B) $x = 4$ , $x = 8$
C) $x = - 8$ , $x = 4$
D) $x = - 10$ , $x = 4$
E) $x = 0$ , $x = 3$

Question 27

Solve the equation by completing the square. $4 x ^ { 2 } = 20 - 32 x$

A) $x = 32$ , $x = - \sqrt { 6 }$
B) $x = - 4 \pm \sqrt { 21 }$
C) $x = 6 \pm \sqrt { 6 }$
D) $x = 4 \pm \sqrt { 21 }$
E) $x = 4$ , $x = - 4$

Question 28

Use the square root method to solve the equation. $x ^ { 2 } = 16$

A) $\pm 4$
B) $1$
C) $4$
D) $- 1$
E) $- 4$

Question 29

Refer to the illustration. Find $\text { c }$ if $a = 4$ and $$b = 3$$ .

A)5
B) $\sqrt { 12 }$
C) $\sqrt { 8 }$
D) $\sqrt { 7 }$
E)7

Question 30

The distance (in feet)that an object will fall in $t$ seconds is given by the formula $s = 16 t ^ { 2 }$ . How long will it take an object to fall 576 feet?

A)6 sec
B)36 sec
C)8 sec
D)7 sec
E)35 sec

Question 31

For this problem, use the triangle in the figure below. Express your two answers in simplest radical form. If a = 8 ft, find b and c .

A) $b = 3 \sqrt { 3 } \mathrm { ft } , c = 6 \sqrt { 3 } \mathrm { ft }$
B) $b = 3 \mathrm { ft } , c = 6 \sqrt { 3 } \mathrm { ft }$
C) $b = 3 \sqrt { 3 } \mathrm { ft } , c = 3 \sqrt { 3 } \mathrm { ft }$
D) $b = 6 \mathrm { ft } , c = 6 \mathrm { ft }$
E) $b = 3 \sqrt { 3 } \mathrm { ft } , c = 6 \mathrm { ft }$

Question 32

When a wholesaler sells $n$ CD players, the revenue is $R = 130 n - \frac { 1 } { 2 } n ^ { 2 }$ . How many CD players would the wholesaler have to sell to receive $\$ 8,450$ ?

A) $100$ CD players
B) $90$ CD players
C) $130$ CD players
D) $150$ CD players
E) $140$ CD players

Question 33

Use the square root method to solve the equation. $3 x ^ { 2 } = 48$

A) $\pm 4$
B) $8$
C) $\pm 8$
D) $- 4$
E) $4$

Question 34

Solve the equation. $6 ( x + 1 ) ^ { 2 } = ( x + 1 ) ^ { 2 } + 40$

A) $1 \pm 2 \sqrt { 2 } ( - 1.8,3.8 )$
B) $- 1 + 2 \sqrt { 2 } ( 1.8 )$
C) $1 - 2 \sqrt { 2 } ( 3.8 )$
D) $- 1 \pm 2 \sqrt { 2 } ( - 3.8,1.8 )$
E) $- 1 - 2 \sqrt { 2 } ( - 3.8 )$

Question 35

Refer to the illustration. Find $b$ if $a = 8$ and $c = 10$ .

A)2
B) $\sqrt { 11 }$
C) $\sqrt { 8 }$
D)6
E) $\sqrt { 164 }$

Question 36

Use the discriminant to determine what type of solutions exist for the quadratic equation. Do not solve the equation. $$5 z ^ { 2 } + 6 z - 3 = 0$$

A)nonreal complex numbers, complex conjugates
B)rational, unequal
C)irrational, unequal
D)rational, equal
E)irrational, equal

Question 37

Use the discriminant to determine what type of solutions exist for the quadratic equation. Do not solve the equation. $3 z ^ { 2 } - 8 z - 3 = 0$

A)nonreal complex numbers, complex conjugates
B)rational, unequal
C)rational, equal
D)irrational, unequal

Question 38

Use the discriminant to determine what type of solutions exist for the quadratic equation. Do not solve the equation. $$4 y ^ { 2 } + y + 5 = 0$$

A)rational, equal
B)irrational, unequal
C)irrational, equal
D)nonreal complex numbers, complex conjugates
E)rational, unequal

Question 39

Use the quadratic formula to find all real solutions of the equation. $2 x ^ { 2 } + x = 5$

A) $\frac { - 1 \pm \sqrt { 41 } } { 4 }$
B) $\frac { - 1 \pm \sqrt { 41 } } { 2 }$
C) $\frac { 1 \pm \sqrt { 41 } } { 2 }$
D) $\frac { 1 \pm \sqrt { 41 } } { 4 }$
E)no real solutions

Question 40

Solve the quadratic equation. $( n - 9 ) ^ { 2 } = 3$ Use the following property: For any real number a , $x ^ { 2 } = a$ if and only if $x = \sqrt { a }$ or $x = - \sqrt { a }$ .

A) $\sqrt { 3 } - 9$ , $- \sqrt { 3 } - 9$
B) $\sqrt { 3 } + 9$ , $0$
C) $\sqrt { 3 } + 9$ , $\sqrt { 3 } - 9$
D) $18$ , $0$
E) $\sqrt { 3 } + 9$ , $- \sqrt { 3 } + 9$

Question 41

Solve the inequality. $\frac { x + 5 } { x - 9 } > 0$

A) $x \in ( - \infty , 5 ) \cup ( 9 , \infty )$
B) $x \in ( - \infty , - 9 ] \cup [ 5 , \infty )$
C) $x \in ( - \infty , - 5 ) \cup ( 9 , \infty )$
D) $x \in ( - \infty , - 5 ] \cup [ 9 , \infty )$
E) $x \in ( - \infty , 5 ] \cup [ 9 , \infty )$

Question 42

Solve the inequality. $$x ^ { 2 } + 11 x + 30 \geq 0$$

A)
B)
C)
D)
E)

Question 43

Solve the inequality. $x ^ { 2 } + 8 x < - 16$

A) $x \in ( - 4,4 )$
B) $x \in ( - \infty , - 4 )$
C) $x \in ( 4 , \infty )$
D) $x \in ( - \infty , \infty )$
E)no solution

Question 44

Solve the inequality. $\frac { 5 x - 2 } { x } \geq 0$

A) $x \in ( - \infty , 0 ) \cup \left[ \frac { 2 } { 5 } , \infty \right)$
B) $x \in ( - \infty , - 5 ) \cup ( 2 , \infty )$
C) $x \in ( - \infty , 2 ] \cup [ 5 , \infty )$
D) $x \in ( - \infty , 2 ) \cup ( 5 , \infty )$
E) $x \in ( - \infty , - 5 ] \cup [ 2 , \infty )$

Question 45

Solve the inequality. $x ^ { 2 } - 8 x + 12 < 0$

A) $x \in ( - 2,6 )$
B) $x \in ( 4,8 )$
C) $x \in ( - 8,4 )$
D) $x \in ( 2,6 )$
E)no solution

Question 46

Solve the inequality. $\frac { x - 1 } { x - 5 } \leq 3$

A) $x \in ( - \infty , - 7 ) \cup ( - 5 , \infty )$
B) $x \in [ 5,7 ]$
C) $x \in ( - \infty , 5 ) \cup [ 7 , \infty )$
D) $x \in ( - \infty , 5 ] \cup [ 7 , \infty )$
E) $x \in ( 5,7 )$

Question 47

The height in feet $h$ of a rocket in flight is given by the formula $h = - 16 t ^ { 2 } + 224 t$ , where $t$ is the time of the flight in seconds. How long will it take for the rocket to hit the ground?

A)13 sec
B)14 sec
C)16 sec
D)12 sec
E)11 sec