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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 , and c .

( [link] ) 2 y 2 3 y + 10 = 0

a = 2 , b = 3 , c = 10

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( [link] ) 10 b 2 = 3 b

a = 10 , b = 3 , c = 0

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

( [link] ) ( 3 x + 5 ) ( x 1 ) = 0

x = 5 3 , 1

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( [link] ) 3 b ( 2 b 1 ) = 0

b = 0 , 1 2

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( [link] ) ( a 8 ) 2 = 0

a = 8

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

( [link] ) 4 x 2 16 = 0

x = 2 , 2

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( [link] ) y 2 12 y + 32 = 0

y = 4 , 8

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( [link] ) a 2 5 a = 14

2 , 7

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( [link] ) 6 a 2 = 10 11 a

a = 5 2 , 2 3

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( [link] ) 2 x 2 = 2 5 x

x = 2 , 1 2

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( [link] ) x 3 25 x = 0

x = 0 , 5 , 5

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

( [link] ) x 2 = 15

x = 15 , 15

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( [link] ) 3 a 2 18 = 0

a = 6 , 6

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( [link] ) ( x 5 ) 2 = 1

x = 4 , 6

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( [link] ) ( y + 11 ) 2 9 = 0

y = 8 , 14

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( [link] ) y 2 25 z 2 = 0 for y

y = 5 z , 5 z

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( [link] ) 6 a 2 18 b 2 c 2 for a

a = ± b c 3

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

( [link] ) x 2 6 x 16 = 0

x = 2 , 8

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( [link] ) y 2 2 y 7 = 0

y = 1 ± 2 2

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( [link] ) ( m + 2 ) 2 5 = 0

m = 2 ± 5

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( [link] ) ( x + b ) 2 = c 2

x = b ± c

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( [link] ) ( x + 1 ) ( x + 4 ) = 6

x = 5 ± 33 2

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( [link] ) 5 z 2 5 z 5 = 2 z 2 z

z = 2 ± 19 3

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( [link] ) 2 m 2 = 5 m

m = 0 , 5 2

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

( [link] ) x 2 + 6 x 8 = 0

x = 3 ± 17

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( [link] ) 2 x 2 + 7 x 12 = 0

x = 7 ± 145 4

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

11 and 13 or 11 and 13

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( [link] ) A study of the air quality by an environmental group suggests that 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 + 0.1 t + 3.1

where 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?

(a) 3.1   (b) 6 years from now

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( [link] ) 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.

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

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

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

( [link] )

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

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

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( [link] )

A graph of a quadratic equation passing through three points with coordinates negative four, negative three;negative three, negative two; and negative two, negtaive three.

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

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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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