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Quadratic Equations: Proficiency Exam

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

Summary: This module is from Elementary Algebra</link> by Denny Burzynski and Wade Ellis, Jr. Methods of solving quadratic equations as well as the logic underlying each method are discussed. Factoring, extraction of roots, completing the square, and the quadratic formula are carefully developed. The zero-factor property of real numbers is reintroduced. The chapter also includes graphs of quadratic equations based on the standard parabola, y = x^2, and applied problems from the areas of manufacturing, population, physics, geometry, mathematics (numbers and volumes), and astronomy, which are solved using the five-step method. This module contains the proficiency exam for the chapter "Quadratic Equations".

Proficiency Exam

For the quadratic equations in the following problems, specify the values of a,b, a,b, and c. c.

Exercise 1

((Reference)) 2 y 2 −3y+10=0 2 y 2 −3y+10=0

Solution

a=2,b=−3,c=10 a=2,b=−3,c=10

Exercise 2

((Reference)) 10 b 2 =3b 10 b 2 =3b

Solution

a=10,b=−3,c=0 a=10,b=−3,c=0

For the following problems, use the zero-factor property to solve each quadratic equation.

Exercise 3

((Reference)) ( 3x+5 )( x−1 )=0 ( 3x+5 )( x−1 )=0

Solution

x=− 5 3 ,1 x=− 5 3 ,1

Exercise 4

((Reference)) 3b( 2b−1 )=0 3b( 2b−1 )=0

Solution

b=0, 1 2 b=0, 1 2

Exercise 5

((Reference)) ( a−8 ) 2 =0 ( a−8 ) 2 =0

Solution

a=8 a=8

For the following problems, solve each quadratic equation by factoring.

Exercise 6

((Reference)) 4 x 2 −16=0 4 x 2 −16=0

Solution

x=−2,2 x=−2,2

Exercise 7

((Reference)) y 2 −12y+32=0 y 2 −12y+32=0

Solution

y=4,8 y=4,8

Exercise 8

((Reference)) a 2 −5a=14 a 2 −5a=14

Solution

−2,7 −2,7

Exercise 9

((Reference)) 6 a 2 =10−11a 6 a 2 =10−11a

Solution

a=− 5 2 , 2 3 a=− 5 2 , 2 3

Exercise 10

((Reference)) 2 x 2 =−2−5x 2 x 2 =−2−5x

Solution

x=−2,− 1 2 x=−2,− 1 2

Exercise 11

((Reference)) x 3 −25x=0 x 3 −25x=0

Solution

x=0,−5,5 x=0,−5,5

For the following problems, solve each quadratic equation by extraction of roots.

Exercise 12

((Reference)) c 2 =81 c 2 =81

Solution

c=−9,9 c=−9,9

Exercise 13

((Reference)) x 2 =15 x 2 =15

Solution

x=− 15 , 15 x=− 15 , 15

Exercise 14

((Reference)) 3 a 2 −18=0 3 a 2 −18=0

Solution

a=− 6 , 6 a=− 6 , 6

Exercise 15

((Reference)) ( x−5 ) 2 =1 ( x−5 ) 2 =1

Solution

x=4,6 x=4,6

Exercise 16

((Reference)) ( y+11 ) 2 −9=0 ( y+11 ) 2 −9=0

Solution

y=−8,−14 y=−8,−14

Exercise 17

((Reference)) y 2 −25 z 2 =0 y 2 −25 z 2 =0 for y y

Solution

y=−5z,5z y=−5z,5z

Exercise 18

((Reference)) 6 a 2 −18 b 2 c 2 6 a 2 −18 b 2 c 2 for a a

Solution

a=±bc 3 a=±bc 3

For the following problems, solve each quadratic equation using quadratic formula.

Exercise 19

((Reference)) x 2 −6x−16=0 x 2 −6x−16=0

Solution

x=−2,8 x=−2,8

Exercise 20

((Reference)) y 2 −2y−7=0 y 2 −2y−7=0

Solution

y=1±2 2 y=1±2 2

Exercise 21

((Reference)) ( m+2 ) 2 −5=0 ( m+2 ) 2 −5=0

Solution

m=−2± 5 m=−2± 5

Exercise 22

((Reference)) ( x+b ) 2 = c 2 ( x+b ) 2 = c 2

Solution

x=−b±c x=−b±c

Exercise 23

((Reference)) ( x+1 )( x+4 )=6 ( x+1 )( x+4 )=6

Solution

x= −5± 33 2 x= −5± 33 2

Exercise 24

((Reference)) 5 z 2 −5z−5=2 z 2 −z 5 z 2 −5z−5=2 z 2 −z

Solution

z= 2± 19 3 z= 2± 19 3

Exercise 25

((Reference)) 2 m 2 =5m 2 m 2 =5m

Solution

m=0, 5 2 m=0, 5 2

For the following problems, solve each quadratic equation by completing the square.

Exercise 26

((Reference)) x 2 +6x−8=0 x 2 +6x−8=0

Solution

x=−3± 17 x=−3± 17

Exercise 27

((Reference)) 2 x 2 +7x−12=0 2 x 2 +7x−12=0

Solution

x= −7± 145 4 x= −7± 145 4

Exercise 28

((Reference)) The product of two consecutive odd integers is 143. What are they?

Solution

11 and 13 or −11 −11 and −13 −13

Exercise 29

((Reference)) A study of the air quality by an environmental group suggests that t t years from now the level of carbon monoxide in the air, in parts per million, will be given by the quadratic equation

 A=0.4 t 2 +0.1t+3.1 A=0.4 t 2 +0.1t+3.1

where A A represents the amount of carbon monoxide in the air.
(a) What is the level, in parts per million, of carbon monoxide in the air now?
(b) How many years from now will the level of carbon monoxide be at18.1 parts per million?

Solution

(a) 3.1 3.1   (b) 6 years from now

Exercise 30

((Reference)) The length of a rectangle is 6 inches longer than the width of the rectangle. Find the dimensions of the rectangle if the area is 112 square feet.

Solution

width= −1+ 1793 4 ; length= 1+ 1793 4 width= −1+ 1793 4 ; length= 1+ 1793 4

For the following problems, construct the graphs of the following equations.

Exercise 31

((Reference)) y= x 2 −3 y= x 2 −3

An xy coordinate plane with gridlines, labeled negative five and five with increments of one units on both axes.

Solution

A graph of a parabola passing through seven points with coordinates negative three, six; negative two, one; negative one, negative two; zero, negative three; one, negative two; two, one; and three, six.

Exercise 32

((Reference)) y= ( x+1 ) 2 y= ( x+1 ) 2

An xy coordinate plane with gridlines, labeled negative five and five with increments of one units on both axes.

Solution

A graph of a parabola passing through five points with coordinates negative three, four; negative two, one; negative one, zero; zero, one; and one, four.

Exercise 33

((Reference)) y= ( x−2 ) 2 +3 y= ( x−2 ) 2 +3

An xy coordinate plane with gridlines, labeled negative five and five with increments of one units on both axes.

Solution

A graph of a parabola passing through three points with coordinates one, four; two, three; and three, four.

For the following problems, write the equation that corresponds to each graph.

Exercise 34

Solution

y= ( x−2 ) 2 +1 or y= x 2 −4x+5 y= ( x−2 ) 2 +1 or y= x 2 −4x+5

Exercise 35

Solution

y=− ( x+3 ) 2 −2 or y=− x 2 −6x−11 y=− ( x+3 ) 2 −2 or y=− x 2 −6x−11

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