Deck 11: Quadratic Equations and Inequalities

Question 1

To multiply factors in the form $\sqrt { - b }$ , rewrite $\sqrt { - b }$ in the form $i \sqrt { b }$ .

Answer 1

True

Question 2

Add. (5 + 7 i ) + (4 + 5 i )

A) 9 + 12 i
B) 12 + 9 i
C) $-$ 9 $-$ 12 i
D) 21

Answer 2

9 + 12 i

Question 3

Write $\frac { \sqrt { - 108 } } { \sqrt { 3 } }$ in terms of i , perform the indicated operations, and simplify.

A) $-$ 6 i
B) 6
C) 6 i

Answer 3

6 i

Question 4

Write $- 3 \sqrt { - 147 }$ in terms of i and simplify.

A) $21 i \sqrt { 3 }$
B) $- 21 \sqrt { 3 }$
C) $- 21 i \sqrt { 3 }$

Question 5

Write $\sqrt { - 294 }$ in terms of i and simplify.

A) $- 7 i \sqrt { 6 }$
B) $7 i \sqrt { 6 }$
C) $7 \sqrt { 6 }$

Question 6

Find ( 2 i ) ( 4 i ) and express the answer in the standard form of a complex number.

A) 2 + 4 i
B) $-$ 8 + 0 i
C) $-$ 8 i
D) 8 + 0 i

Question 7

Write $\frac { \sqrt { - 169 } } { \sqrt { - 16 } }$ in terms of i , perform the indicated operations, and simplify.

A) $\frac { 13 } { 4 } i$
B) $- \frac { 13 } { 4 }$
C) $\frac { 13 } { 4 }$

Question 8

Write $\sqrt { - 16 }$ in terms of i and simplify.

A) $-$ 4 i
B) $-$ 4
C) 4 i

Question 9

Write $\frac { \sqrt { - 160 } } { \sqrt { - 10 } }$ in terms of i , perform the indicated operations, and simplify.

A) $-$ 4
B) 4
C) 4 i

Question 10

Add or subtract as indicated. (6 + 8 i ) $-$ (4 + 6 i )

A) 2 + 2 i
B) 14 + 10 i
C) $-$ 2 $-$ 2 i
D) $-$ 10 $-$ 14 i

Question 11

Write $- 3 \sqrt { - 45 }$ in terms of i and simplify.

A) $- 9 i \sqrt { 5 }$
B) $9 i \sqrt { 5 }$
C) $- 9 \sqrt { 5 }$

Question 12

Add or subtract as indicated. ( $-$ 3 $-$ 4 i ) $-$ ( $-$ 6 $-$ 7 i )

A) 11 + 9 i
B) $-$ 3 + 11 i
C) $-$ 9 $-$ 3 i
D) 3 + 3 i

Question 13

Add or subtract as indicated. ( $-$ 3 $-$ 2 i ) + (8 $-$ 7 i )

A) $-$ 11 $-$ 9 i
B) 9 + 11 i
C) 5 $-$ 9 i
D) $-$ 5 + 9 i

Question 14

The set of complex numbers is a subset of the set of imaginary numbers.

Question 15

For the complex number $a + b i$ , the a is the real part of the complex number, and bi is called the imaginary part.

Question 16

Write $\sqrt { 6 } \sqrt { - 25 }$ in terms of i , perform the indicated operations, and simplify.

A) $- 5 i \sqrt { 6 }$
B) $5 i \sqrt { 6 }$
C) $5 \sqrt { 6 }$

Question 17

The product of a complex number and its conjugate is always a real number.

Question 18

Write $\sqrt { - 2 } \sqrt { - 4 }$ in terms of i , perform the indicated operations, and simplify.

A) $- 2 \sqrt { 2 }$
B) $- 2 i \sqrt { 2 }$
C) $2 \sqrt { 2 }$

Question 19

Write $\sqrt { - \frac { 25 } { 16 } }$ in terms of i and simplify.

A) $\frac { 5 } { 4 }$
B) $\frac { 5 } { 4 } i$
C) $- \frac { 5 } { 4 } i$

Question 20

Write $\sqrt { - 64 } \sqrt { - 49 }$ in terms of i , perform the indicated operations, and simplify.

A) 56 i
B) 56
C) $-$ 56

Question 21

Add or subtract as indicated. ( 3 5 i ) ( 6 7 i )

Question 22

Write in terms of i , perform the indicated operations, and simplify.

Question 23

Add or subtract as indicated. (7 + 7 i ) (4 + 3 i )

Question 24

Write in terms of i , perform the indicated operations, and simplify.

Question 25

Write in terms of i and simplify.

Question 26

Find ( $-$ 5 i ) ( $-$ 5 $-$ 2 i ) and express the answer in the standard form of a complex number.

A) $-$ 10 i $-$ 25
B) $-$ 25 i + 10
C) 25 i $-$ 10
D) 10 i $-$ 25

Question 27

Find the quotient $\frac { - 6 + 18 i } { - 1 + i }$ and express the answer in the standard form of a complex number.

A) 12 i $-$ 6
B) 24 $-$ 12 i
C) 12 $-$ 6 i
D) $-$ 12 + 6 i

Question 28

Find $( 2 + 7 i ) ( 4 + 3 i )$ and express the answer in the standard form of a complex number.

A) $8 + 21 i$
B) $- 29 - 34 i$
C) $- 13 + 34 i$
D) $29 + 34 i$

Question 29

Write in terms of i , perform the indicated operations, and simplify.

Question 30

Find $( 7 + 4 i ) ^ { 2 }$ and express the answer in the standard form of a complex number.

A) 33 $-$ 56 i
B) $-$ 33 $-$ 56 i
C) 33 i + 56
D) 33 + 56 i

Question 31

Find the quotient $\frac { 5 } { 13 i }$ and express the answer in the standard form of a complex number.

A) $- \frac { 5 } { 13 }$
B) $- \frac { 5 } { 13 } i$
C) $- \frac { 13 } { 5 } i$
D) $\frac { 5 } { 13 } i$

Question 32

Write in terms of i and simplify.

Question 33

Add or subtract as indicated. ( 3 9 i ) + (4 6 i )

Question 34

Find $( 8 - 5 i ) ( - 2 - i )$ and express the answer in the standard form of a complex number.

A) $- 16 - 18 i$
B) $21 - 2 i$
C) $- 11 + 2 i$
D) $- 21 + 2 i$

Question 35

Write in terms of i , perform the indicated operations, and simplify.

Question 36

Write in terms of i and simplify.

Question 37

Write in terms of i , perform the indicated operations, and simplify. Please enter your answer as a fraction.

Question 38

Find $( 3 + 9 i ) ( - 4 - 5 i )$ and express the answer in the standard form of a complex number.

A) $- 57 - 36 i$
B) $33 + 51 i$
C) $- 12 - 45 i$
D) $33 - 51 i$

Question 39

Add. (7 + 8 i ) + (5 + 2 i )

Question 40

Write in terms of i and simplify.

Question 41

Solve the quadratic equation by factoring and applying the property ab = 0 if and only if a = 0 or b = 0. x 2 $-$ 2 x = 0

A) $x = \sqrt { 2 }$
B) $x = 0 \text { or } x = \sqrt { 2 }$
C) x = 2
D) x = 0 or x = 2

Question 42

Write each of the following in terms of i and simplify. Match each expression with the corresponding answer.

- $7 i \sqrt { 3 }$

A) $\sqrt { - 147 }$
B) $- 2 \sqrt { - 96 }$

Question 43

Find ( 6 i ) ( 9 i ) and express the answer in the standard form of a complex number.

Question 44

Find and express the answer in the standard form of a complex number.

Question 45

The standard form of a quadratic equation is $a x ^ { 2 } + b x + c = 0$ , where $a \neq 0$ .

Question 46

Find each of the products and match each expression with the corresponding answer.

- $60 - 32 i$

A) $( 8 + 2 i ) ^ { 2 }$
B) $( 8 - 2 i ) ^ { 2 }$

Question 47

Find the quotient and express the answer in the standard form of a complex number.

Question 48

Solve the quadratic equation by factoring and applying the property ab = 0 if and only if a = 0 or b = 0. x 2 $-$ 4 x $-$ 32 = 0

A) x = 4 or x = 8
B) x = $-$ 4 or x = 8
C) x = $-$ 4 or x = $-$ 8
D) x = 4 or x = $-$ 8

Question 49

In the complex number system there is no solution to the equation $x ^ { 2 } = - 6$ .

Question 50

Write each of the following in terms of i and simplify. Match each expression with the corresponding answer.

- $- 8 i \sqrt { 6 }$

A) $\sqrt { - 147 }$
B) $- 2 \sqrt { - 96 }$

Question 51

Find and express the answer in the standard form of a complex number.

Question 52

Find and express the answer in the standard form of a complex number.

Question 53

Find ( 3 i ) ( 9 2 i ) and express the answer in the standard form of a complex number.

Question 54

Find each of the products and match each expression with the corresponding answer.

- $60 + 32 i$

A) $( 8 + 2 i ) ^ { 2 }$
B) $( 8 - 2 i ) ^ { 2 }$

Question 55

Write in terms of i , perform the indicated operations, and simplify.

Question 56

The equation $\sqrt { x } = x ^ { 2 } - 4$ is quadratic in form.

Question 57

Find the quotient and express the answer in the standard form of a complex number. Please enter your answer as a fraction.

Question 58

In a $30 ^ { \circ } - 60 ^ { \circ }$ triangle, the hypotenuse is twice as long in length as the side opposite the $30 ^ { \circ }$ angle.

Question 59

Find and express the answer in the standard form of a complex number.

Question 60

Solve the quadratic equation by factoring and applying the property " ab = 0 if and only if a = 0 or b = 0." 3 x ² + 28 x + 32 = 0

A) $x = - 8 \text { or } x = - \frac { 4 } { 3 }$
B) x = $-$ 8 or x = $-$ 4
C) x = $-$ 8 or x = 4
D) x = 8 or x = 3

Question 61

Solve the radical equation. Don t forget, you must check potential solutions. $5 \sqrt { x } = x + 4$

A) x = 16
B) x = 1
C) x = 1 or x = 16
D) x = $-$ 1 or x = 16

Question 62

Solve the quadratic equation. ( x $-$ 5) 2 = 64

A) x = 13 or x = -3
B) x = 13 or x = -13
C) x = 8 or x = -8
D) x = -3 or x = 3

Question 63

Solve the quadratic equation. ( n $-$ 2) 2 = 7

A) $n = \sqrt { 7 } - 2 \text { or } n = - \sqrt { 7 } - 2$
B) $n = \sqrt { 7 } + 2 \text { or } n = \sqrt { 7 } - 2$
C) n = 51 or n = 47
D) $n = \sqrt { 7 } + 2 \text { or } n = - \sqrt { 7 } + 2$

Question 64

For this problem a and b represent the lengths of the legs of a right triangle, and c represents the length of the hypotenuse. Find c if a = 10 centimeters and b = 15 centimeters. Express the answer in simplest radical form.

A) $\sqrt { 25 }$ centimeters
B) 25 centimeters
C) $5 \sqrt { 13 }$ centimeters
D) $5 \sqrt { 5 }$ centimeters

Question 65

Solve the radical equation. Don t forget, you must check potential solutions. $\sqrt { 5 x } + 10 = x$

A) x = 5
B) x = 20
C) x = 20 or x = 5
D) x = 20 or x = $-$ 5

Question 66

Solve the quadratic equation. n 2 $-$ 5 = 0

A) $n = \sqrt { 5 } \text { or } n = - \sqrt { 5 }$
B) n = 5 or n = 0
C) n = $-$ 5
D) $n = \sqrt { 5 }$

Question 67

Solve the quadratic equation. 4( x + 3) 2 $-$ 4 = 12

A) x = -1 or x = -5
B) x = 6 or x = 2
C) x = 4 or x = -8
D) x = -5 or x = -4

Question 68

Solve the quadratic equation. $12 n ^ { 2 } = 15$

A) $x = - \frac { \sqrt { 5 } } { 2 } \text { or } x = \frac { \sqrt { 5 } } { 2 }$
B) $x = - \frac { \sqrt { 2 } } { 5 } \text { or } x = \frac { \sqrt { 2 } } { 5 }$
C) $x = - \frac { \sqrt { 10 } } { 2 } \text { or } x = \frac { \sqrt { 10 } } { 2 }$
D) $x = - \frac { \sqrt { 5 } } { 10 } \text { or } x = \frac { \sqrt { 5 } } { 10 }$

Question 69

Solve the quadratic equation. $8 t ^ { 2 } + 15 = 0$

A) $t = - \frac { \sqrt { 30 } } { 4 } i \text { or } t = \frac { \sqrt { 30 } } { 4 } i$
B) $t = - \frac { \sqrt { 15 } } { 8 } i \text { or } t = \frac { \sqrt { 15 } } { 8 } i$
C) $t = - \frac { \sqrt { 30 } } { 4 } \text { or } t = \frac { \sqrt { 30 } } { 4 }$
D) $t = - \frac { \sqrt { 15 } } { 2 } i \text { or } t = \frac { \sqrt { 15 } } { 2 } i$

Question 70

Solve the quadratic equation. x ² = 16

A) x = 4 or x = 0
B) x = 4
C) x = 4 or x = $-$ 4
D) x = 4 or x = 16

Question 71

For this problem use the triangle in the figure. If a = 7 feet find b and c . Express the answers in simplest radical form.

A) $b = 7 \sqrt { 3 } \text { feet, } c = 14 \sqrt { 3 } \text { feet }$
B) $b = 14 \text { feet, } c = 14 \text { feet }$
C) $b = 7 \sqrt { 3 } \text { feet, } c = 14 \text { feet }$
D) $b = 7 \sqrt { 3 } \text { feet, } c = 7 \sqrt { 3 } \text { feet }$

Question 72

Solve the quadratic equation by factoring and applying the property " ab = 0 if and only if a = 0 or b = 0." x ² + 9 kx = 0

A) x = $-$ 9 k or x = 0
B) k = $-$ 9
C) x = 9 k
D) x = 9 or x = k

Question 73

A 83-foot guy-wire makes an angle of 60 o ; with the ground and is attached to a telephone pole (see the figure). Find the distance from the base of the pole to the point on the pole where the wire is attached. Express your answer to the nearest tenth of a foot.

A) 71.9 feet
B) 69.6 feet
C) 77.5 feet
D) 73.1 feet

Question 74

Solve the quadratic equation. (4 x + 1) ² = 16

A) $n = \frac { 3 } { 4 } \text { or } n = - \frac { 5 } { 4 }$
B) x = 3 or x = $-$ 3
C) $n = \frac { 3 } { 4 } \text { or } n = - \frac { 3 } { 4 }$
D) x = 3 or x = 5

Question 75

Solve the radical equation. Don t forget, you must check potential solutions. x = __________

Question 76

Solve the quadratic equation by factoring and applying the property ab = 0 if and only if a = 0 or b = 0. x 2 + 2 kx $-$ 8 k 2 = 0

A) x = $-$ 2 k or x = $-$ 4
B) x = $-$ 3 or x = 5 k
C) x = 2 k or x = $-$ 4 k
D) x = 8 or x = k

Question 77

For this problem a and b represent the lengths of the legs of a right triangle, and c represents the length of the hypotenuse. Find a if $c = 14$ and $b = 5$ inches. Express the answer in simplest radical form.

A) $\sqrt { 221 }$ inches
B) $- \sqrt { 221 }$ inches
C) $- 3 \sqrt { 19 }$ inches
D) $3 \sqrt { 19 }$ inches

Question 78

For this problem use the isosceles right triangle in the figure. If c = 16 meters find a and b . Express the answers in simplest radical form.

A) $a = 8 \sqrt { 2 } \text { meters, } b = 16 \sqrt { 2 } \text { meters }$
B) $a = 16 \sqrt { 2 } \text { meters, } b = 16 \sqrt { 2 } \text { meters }$
C) $a = 8 \text { meters, } b = 8 \text { meters }$
D) $a = 8 \sqrt { 2 } \text { meters, } b = 8 \sqrt { 2 } \text { meters }$

Question 79

Solve the quadratic equation. 4 t ² = 32

A) $n = \sqrt { 2 } \text { or } n = - \sqrt { 2 }$
B) n = 4 or n = $-$ 4
C) $n = 2 \sqrt { 2 } \text { or } n = - 2 \sqrt { 2 }$
D) n = 2 or n = $-$ 2

Question 80

A diagonal of a square parking lot is 55 meters. Find, to the nearest meter, the length of a side of the lot.

A) 34 meters
B) 36 meters
C) 39 meters
D) 37 meters