Deck 9: Quadratic Equations

Question 1

Solve the equation by completing the square. $2 x ^ { 2 } = 4 - 8 x$

A) $x = 8$ , $x = - \sqrt { 2 }$
B) $x = - 2 \pm \sqrt { 6 }$
C) $x = 3 \pm \sqrt { 2 }$
D) $x = 2 \pm + \sqrt { 6 }$
E) $x = 2$ , $x = - 2$

Answer 1

$x = - 2 \pm \sqrt { 6 }$

Question 2

Solve the equation by completing the square. $x ^ { 2 } - 8 x - 9 = 0$

A) $x = - 2$ , $x = 9$
B) $x = 8$ , $x = 9$
C) $x = 1$ , $x = 9$
D) $x = - 1$ , $x = 9$
E) $x = - 8$ , $x = 2$

Answer 2

$x = - 1$ , $x = 9$

Question 3

Use the quadratic formula to find all real solutions of the equation. $x ^ { 2 } - 7 x + 10 = 0$

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

Answer 3

x = 5, x = 2

Question 4

Use the method of completing the square to solve the quadratic equation. $2 x ^ { 2 } + 10 x - 9 = 0$

A) $x = \frac { - 10 \pm \sqrt { 43 } } { 2 }$
B) $x = \frac { \pm 10 \pm \sqrt { 43 } } { 2 }$
C) $x = \frac { \pm 5 \pm \sqrt { 43 } } { 2 }$
D) $x = \frac { 10 \pm \sqrt { 43 } } { 2 }$
E) $x = \frac { - 5 \pm \sqrt { 43 } } { 2 }$

Question 5

When a wholesaler sells n CD players, the revenue is $R = 140 n - \frac { 1 } { 2 } n ^ { 2 }$ . How many CD players would the wholesaler have to sell to receive $\$ 9,800$ ?

A) $170$ CD players
B) $180$ CD players
C) $140$ CD players
D) $160$ CD players
E) $150$ CD players

Question 6

Use the quadratic formula to find all real solutions of the equation. $2 x ^ { 2 } - 3 x = 4$

A) $\frac { 3 \pm \sqrt { 41 } } { 4 }$
B) $\frac { 3 \pm \sqrt { 41 } } { 2 }$
C) $\frac { - 3 \pm \sqrt { 41 } } { 2 }$
D) $\frac { - 3 \pm \sqrt { 41 } } { 4 }$
E)no real solutions

Question 7

Solve the equation. $5 ( x + 2 ) ^ { 2 } = ( x + 2 ) ^ { 2 } + 32$

A) $2 \pm 2 \times \sqrt { 2 } ( - 0.8,4.8 )$
B) $- 2 + 2 \sqrt { 2 } ( 0.8 )$
C) $2 - 2 \sqrt { 2 } ( 4.8 )$
D) $- 2 \pm 2 \sqrt { 2 } ( - 4.8,0.8 )$
E) $- 2 - 2 \sqrt { 2 } ( - 4.8 )$

Question 8

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

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

Question 9

Solve the equation. Give the solutions rounded to the nearest hundredth. x ² + 12 x + 17 = 0

A)
B)
C)
D)
E)

Question 10

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 784 feet?

A)7 sec
B)49 sec
C)9 sec
D)6 sec
E)48 sec

Question 11

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 12

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

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

Question 13

Solve the equation by completing the square. $x ^ { 2 } = 14 - 5 x$

A) $x = 1$ , $x = 10$
B) $x = 2$ , $x = 7$
C) $x = - 7$ , $x = 2$
D) $x = - 10$ , $x = 2$
E) $x = 0$ , $x = 1$

Question 14

Complete the square and factor the resulting perfect-square trinomial. $8 t ^ { 2 } + 120 t$

A)( 8 t + $\frac { 225 } { 4 }$ ) $2$
B)( 8 t - $\frac { 15 } { 2 }$ ) $2$
C)( 8 t + $\frac { 15 } { 2 }$ ) $2$
D)( 8 t - $\frac { 225 } { 4 }$ ) $2$

Question 15

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 16

The height in feet h of a rocket in flight is given by the formula $h = - 16 t ^ { 2 } + 160 t$ , where t is the time of the flight in seconds. How long will it take for the rocket to hit the ground?

A)11 sec
B)10 sec
C)8 sec
D)12 sec
E)13 sec

Question 17

Complete the square and factor the resulting perfect-square trinomial. $t ^ { 2 } + 8 t$

A)( t + 16 ) $2$
B)( t - 4 ) $2$
C)( t + 4 ) $2$
D)( t - 16 ) $2$

Question 18

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

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

Question 19

What is the first step to solving x ² 10 x = 56 by completing the square?

A)Add 25 to both sides.
B)Add -25 to both sides
C)Subtract 56 from both sides.
D)Isolate x ² .
E)none of these

Question 20

Use the method of completing the square to solve the quadratic equation. $3 x ^ { 3 } + 7 x - 1 = 0$

A) $x = \frac { - 7 + \sqrt { 61 } } { 2 }$
B) $x = \frac { - 7 \pm \sqrt { 61 } } { 6 }$
C) $x = \frac { \pm 7 \pm \sqrt { 61 } } { 2 }$
D) $x = \frac { \pm 7 + \sqrt { 61 } } { 2 }$
E) $x = \frac { \pm 7 \pm \sqrt { 61 } } { 6 }$

Question 21

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

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

Question 22

The number of cases of flu seen by doctors at a county health clinic each week during a 10-week period is described by the quadratic function f ( x )= -4 x ² + 40 x (graphed in the illustration). What point gives us information of the most cases of flu?

A)(5, 100)
B)(8, 100)
C)(10, 0)
D)(5, 0)

Question 23

What coefficient indicates whether the graph of the parabola opens upward or downward?

Question 24

Solve the following quadratic equation using the method that seems most appropriate to you. $5 x ^ { 2 } + 2 x - 3 = 0$

A) $\left\{ - 1 , \frac { 3 } { 5 } \right\}$
B) $\left\{ - 1 , \frac { 5 } { 3 } \right\}$
C) $\left\{ - \frac { 5 } { 3 } , 1 \right\}$
D) $\left\{ - \frac { 3 } { 5 } , 1 \right\}$
E)No real number solutions

Question 25

Find the vertex of the graph of y = 2 x ² 24 x + 19.

A)(12, 53)
B)(6, -53)
C)(-12, -53)
D)(-6, -53)
E)(-6, 53)

Question 26

Solve the following quadratic equation using the method that seems most appropriate to you. a $2$ + 3 a = 10

A){ -5, 2 }
B){ -9, 2 }
C){ -9, 6 }
D){ -5, 6 }
E)No real number solutions

Question 27

What is the 4th step in graphing a quadratic equation $y = a x ^ { 2 } + b x + c$ ?

A)Find the intercepts.
B)Plot points, using symmetry.
C)Draw the parabola.
D)Test for opening upward\downward.

Question 28

Look at the graph of the equation $9 x ^ { 2 } - 42 x + 59$ . How many solutions does the equation have? What are they?

A)3 solutions, $\frac { 7 } { 3 }$ , $- \frac { 7 } { 3 }$ , and 0
B)no solutions
C)2 solutions, $\frac { 7 } { 3 }$ and $- \frac { 7 } { 3 }$
D)1 repeated solution, $\frac { 7 } { 3 }$

Question 29

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

A) $7 i \sqrt { 6 }$
B) $- 7 i \sqrt { 6 }$
C) $7 \sqrt { 6 }$
D) $42 i$
E) $- 7 \sqrt { 6 i }$

Question 30

Look at the graph of the equation $25 x ^ { 2 } - 30 x + 9$ . How many solutions does the equation have? What are they?

A)3 solutions, $\frac { 3 } { 5 }$ , $- \frac { 3 } { 5 }$ , and 0
B)no solutions
C)2 solutions, $\frac { 3 } { 5 }$ and $- \frac { 3 } { 5 }$
D)1 repeated solution, $\frac { 3 } { 5 }$

Question 31

The area of a circle is numerically equal to 8 times the circumference of the circle. Find the length of a radius of the circle.

A)16 $\pi$
B)17
C)16
D)15 $\pi$
E)8 $\pi$

Question 32

Refer to the graph of f ( x )= 2 x ² 2 x 4 given below and find the y -intercept. y -intercept = __________

Question 33

How many x -intercepts does the following parabola have?

A)0
B)2
C)1

Question 34

The area of a rectangular region is 187 square feet. If the length and width are each increased by 5 feet, the area is increased by 165 square feet. Find the length and width of the original rectangle.

A)13 feet by 19 feet
B)11 feet by 17 feet
C)10 feet by 16 feet
D)8 feet by 14 feet
E)12 feet by 18 feet

Question 35

Solve the following quadratic equation using the method that seems most appropriate to you. (4 m + 20) $2$ = 256

A){ 3, 9 }
B){ -3, 1 }
C){ -1, 3 }
D){ -9, -1 }
E)No real number solutions

Question 36

Write $\sqrt { - \frac { 49 } { 4 } }$ in terms of i and simplify.

A) $\frac { 7 } { 2 }$
B) $- \frac { 7 } { 2 } + i$
C) $\frac { 7 } { 2 } + i$
D) $- \frac { 7 i } { 2 }$
E) $\frac { 7 i } { 2 }$

Question 37

Graph the parabola $$y = 4 x ^ { 2 } + 5 x$$ .

A)
B)
C)
D)

Question 38

The function y = 16 x ² + 4 x + 10 is written in y = ax ² bx c form. What are a , b and c ?

Question 39

Graph the equation $y = - 0.5 x ^ { 2 } + 1.5 x + 10$ .

A)
B)
C)
D)

Question 40

The area of a tennis court is 2,128 square feet (see illustration). The length of the court is $\frac { 19 } { 7 }$ times the width. Find the length and width of the tennis court.

A)94 feet by 25 feet
B)82 feet by 26 feet
C)85 feet by 25 feet
D)76 feet by 22 feet
E)76 feet by 28 feet

Question 41

Solve the following quadratic equation. $t ^ { 2 } + 5 t + 9 = 0$

A) $\{ 2.5 + i \sqrt { 2.75 } , 2.5 - i \sqrt { 2.75 } \}$
B) $\{ 2.5 + i \sqrt { 3 } , 2.5 - i \sqrt { 3 } \}$
C) $\{ - 2.5 + i \sqrt { 2.75 } , - 2.5 - i \sqrt { 2.75 } \}$
D) $\{ - 5 + i \sqrt { 2.75 } , - 5 - i \sqrt { 2.75 } \}$
E) $\{ - 5 + i \sqrt { - 2.75 } , - 5 - i \sqrt { - 2.75 } \}$

Question 42

Solve the following quadratic equation. $( x - 4 ) ^ { 2 } = - 64$

A) $\{ 8 i , - 8 i \}$
B) $\{ 8 i + 4 , - 8 i + 4 \}$
C) $\{ 4 i + 8 , - 4 i + 8 \}$
D) $\{ 8 i + 4 \}$
E) $\{ 12 , - 12 \}$

Question 43

Write the number 3.9 in the form a + bi.

A)3.9 + 0 i
B)3.9 i - 0
C)3.9
D)3.9 i

Question 44

Write $\sqrt { - 3 } \sqrt { - 64 }$ in terms of i , perform the indicated operations, and simplify.

A) $- 8 \sqrt { 3 }$
B) $- 8 i \sqrt { 3 }$
C) $24 i$
D) $8 i \sqrt { 3 }$
E) $8 \sqrt { 3 }$

Question 45

Simplify the expression. Enter the answer in a + bi form. $( 7 + i ) - ( 1 - 2 i )$

A) $6 + 3 i$
B) $6 - 3 j$
C) $6 + i$
D) $6 - i$

Question 46

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 47

Solve the equation. Enter all roots in bi or a + bi form. $4 x ^ { 2 } + x + 6 = 0$

A) $\frac { - 1 \pm \sqrt { 95 } } { 8 } i$
B) $\frac { - 1 } { 8 } \pm \frac { \sqrt { 95 } } { 8 } i$
C) $\frac { - 1 } { 4 } \pm \frac { \sqrt { 95 } } { 4 } i$
D) $\frac { - 1 \pm \sqrt { 95 } } { 4 } i$

Question 48

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 49

Perform the operation. Write the answer in a + bi form. $( 3 + 2 \sqrt { 2 } i ) + ( - 6 - \sqrt { 2 } i )$

A) $- 3 - 2 \sqrt { 2 } i$
B) $- 2 - \sqrt { 2 } i$
C) $- 3 + \sqrt { 2 } i$
D) $- 9 + 2 \sqrt { 2 } i$

Question 50

Write the number $\sqrt { - 4.9 }$ in the form a + bi.

A)4.9 + 0 i
B)0 + 4.9 i
C)4.9
D)-4.9 i

Question 51

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 52

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

A) $\frac { 30 } { 11 } + \frac { 2 } { 11 } i$
B) $\frac { 1 } { 11 } - 2 \frac { \sqrt { 30 } } { 11 } i$
C) $\frac { 11 } { 30 } + \frac { \sqrt { 30 } } { 11 } i$
D) $\frac { 1 } { 11 } + 2 \frac { \sqrt { 30 } } { 11 } i$
E) $\frac { 11 } { 30 } - \frac { \sqrt { 30 } } { 11 } i$

Question 53

Write the number $\sqrt { - 8.6 } - 9.9$ in the form a + bi.

A)9.9 i - 8.6
B)9.9 i + 8.6
C)9.9 + 8.6 i
D)9.9 - 8.6 i