<< Chapter < Page Chapter >> Page >
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.Objectives of this module: be able to solve quadratic equations by factoring.

Overview

  • Factoring Method
  • Solving Mentally After Factoring

Factoring method

To solve quadratic equations by factoring, we must make use of the zero-factor property.

  1. Set the equation equal to zero, that is, get all the nonzero terms on one side of the equal sign and 0 on the other.

    a x 2 + b x + c = 0
  2. Factor the quadratic expression.

    ( ) ( ) = 0
  3. By the zero-factor property, at least one of the factors must be zero, so, set each of the factors equal to 0 and solve for the variable.

Sample set a

Solve the following quadratic equations. (We will show the check for problem 1.)

x 2 7 x + 12 = 0. The equation is already  set equal to 0 . Factor . ( x 3 ) ( x 4 ) = 0 Set each factor equal to 0 . x 3 = 0 or x 4 = 0 x = 3 or x = 4
C h e c k : If x = 3 , x 2 7 x + 12 = 0 3 2 7 · 3 + 12 = 0 Is this correct? 9 21 + 12 = 0 Is this correct? 0 = 0 Yes, this is correct .

C h e c k : If x = 4 , x 2 7 x + 12 = 0 4 2 7 · 4 + 12 = 0 Is this correct? 16 28 + 12 = 0 Is this correct? 0 = 0 Yes, this is correct .
Thus, the solutions to this equation are x = 3 , 4.

Got questions? Get instant answers now!

x 2 = 25. Set the equation equal to  0. x 2 25 = 0 Factor . ( x + 5 ) ( x 5 ) = 0 Set each factor equal to  0. x + 5 = 0 or x 5 = 0 x = 5 or x = 5
Thus, the solutions to this equation are x = 5 , 5.

Got questions? Get instant answers now!

x 2 = 2 x . Set the equation equal to  0. x 2 2 x = 0 Factor . x ( x 2 ) Set each factor equal to  0. x = 0 or x 2 = 0 x = 2
Thus, the solutions to this equation are x = 0 , 2.

Got questions? Get instant answers now!

2 x 2 + 7 x 15 = 0. Factor . ( 2 x 3 ) ( x + 5 ) = 0 Set each factor equal to  0. 2 x 3 = 0 or x + 5 = 0 2 x = 3 or x = 5 x = 3 2
Thus, the solutions to this equation are x = 3 2 , 5.

Got questions? Get instant answers now!

63 x 2 = 13 x + 6
63 x 2 13 x 6 = 0 ( 9 x + 2 ) ( 7 x 3 ) = 0 9 x + 2 = 0 or 7 x 3 = 0 9 x = 2 or 7 x = 3 x = 2 9 or x = 3 7
Thus, the solutions to this equation are x = 2 9 , 3 7 .

Got questions? Get instant answers now!

Practice set a

Solve the following equations, if possible.

( x 7 ) ( x + 4 ) = 0

x = 7 , 4

Got questions? Get instant answers now!

( 2 x + 5 ) ( 5 x 7 ) = 0

x = 5 2 , 7 5

Got questions? Get instant answers now!

x 2 + 2 x 24 = 0

x = 4 , 6

Got questions? Get instant answers now!

6 x 2 + 13 x 5 = 0

x = 1 3 , 5 2

Got questions? Get instant answers now!

5 y 2 + 2 y = 3

y = 3 5 , 1

Got questions? Get instant answers now!

m ( 2 m 11 ) = 0

m = 0 , 11 2

Got questions? Get instant answers now!

6 p 2 = ( 5 p + 1 )

p = 1 3 , 1 2

Got questions? Get instant answers now!

r 2 49 = 0

r = 7 , 7

Got questions? Get instant answers now!

Solving mentally after factoring

Let’s consider problems 4 and 5 of Sample Set A in more detail. Let’s look particularly at the factorizations ( 2 x 3 ) ( x + 5 ) = 0 and ( 9 x + 2 ) ( 7 x 3 ) = 0. The next step is to set each factor equal to zero and solve. We can solve mentally if we understand how to solve linear equations: we transpose the constant from the variable term and then divide by the coefficient of the variable.

Sample set b

Solve the following equation mentally.

( 2 x 3 ) ( x + 5 ) = 0
2 x 3 = 0 Mentally add 3 to both sides . The constant changes sign . 2 x = 3 Divide by 2, the coefficient of  x . The 2 divides the constant 3 into  3 2 .  The coefficient becomes the denominator . x = 3 2 x + 5 = 0 Mentally subtract 5 from both sides . The constant changes sign . x = 5 Divide by the coefficient of   x , 1 .The coefficient becomes the denominator . x = 5 1 = 5 x = 5
Now, we can immediately write the solution to the equation after factoring by looking at each factor, changing the sign of the constant, then dividing by the coefficient.

Got questions? Get instant answers now!

Practice set b

Solve ( 9 x + 2 ) ( 7 x 3 ) = 0 using this mental method.

x = 2 9 , 3 7

Got questions? Get instant answers now!

Exercises

For the following problems, solve the equations, if possible.

( x + 1 ) ( x + 3 ) = 0

x = 1 , 3

Got questions? Get instant answers now!

( x 5 ) ( x 1 ) = 0

x = 1 , 5

Got questions? Get instant answers now!

( x 6 ) ( x 3 ) = 0

Got questions? Get instant answers now!

( x 4 ) ( x + 2 ) = 0

x = 2 , 4

Got questions? Get instant answers now!

( x + 6 ) ( x 1 ) = 0

Got questions? Get instant answers now!

( 2 x + 1 ) ( x 7 ) = 0

x = 1 2 , 7

Got questions? Get instant answers now!

( 3 x + 2 ) ( x 1 ) = 0

Got questions? Get instant answers now!

( 4 x + 3 ) ( 3 x 2 ) = 0

x = 3 4 , 2 3

Got questions? Get instant answers now!

( 5 x 1 ) ( 4 x + 7 ) = 0

Got questions? Get instant answers now!

( 6 x + 5 ) ( 9 x 4 ) = 0

x = 5 6 , 4 9

Got questions? Get instant answers now!

( 3 a + 1 ) ( 3 a 1 ) = 0

Got questions? Get instant answers now!

x ( x + 4 ) = 0

x = 4 , 0

Got questions? Get instant answers now!

y ( 3 y 4 ) = 0

y = 0 , 4 3

Got questions? Get instant answers now!

x ( 2 x + 1 ) ( 2 x + 8 ) = 0

x = 4 , 1 2 , 0

Got questions? Get instant answers now!

y ( 5 y + 2 ) ( 2 y 1 ) = 0

Got questions? Get instant answers now!

( b + 7 ) 2 = 0

b = 7

Got questions? Get instant answers now!

x ( x 4 ) 2 = 0

x = 0 , 4

Got questions? Get instant answers now!

y ( y 7 ) 2 = 0

y = 0 , 7

Got questions? Get instant answers now!

x 2 4 = 0

x = 2 , 2

Got questions? Get instant answers now!

x 2 + 36 = 0

no solution

Got questions? Get instant answers now!

a 2 100 = 0

a = 10 , 10

Got questions? Get instant answers now!

b 2 49 = 0

b = 7 , 7

Got questions? Get instant answers now!

3 a 2 75 = 0

a = 5 , 5

Got questions? Get instant answers now!

y 3 y = 0

y = 0 , 1 , 1

Got questions? Get instant answers now!

b 2 = 4

b = 2 , 2

Got questions? Get instant answers now!

a 2 = 36

a = 6 , 6

Got questions? Get instant answers now!

2 x 2 = 4

x = 2 , 2

Got questions? Get instant answers now!

7 b 2 = 63

b = 3 , 3

Got questions? Get instant answers now!

3 b 2 = 48

b = 4 , 4

Got questions? Get instant answers now!

y 2 + 10 y + 25 = 0

y = 5

Got questions? Get instant answers now!

x 2 2 x 1 = 0

no solution

Got questions? Get instant answers now!

a 2 + 4 a + 4 = 0

a = 2

Got questions? Get instant answers now!

b 2 14 b = 49

b = 7

Got questions? Get instant answers now!

2 m 3 + 4 m 2 + 2 m = 0

m = 0 , 1

Got questions? Get instant answers now!

3 m n 2 36 m n + 36 m = 0

Got questions? Get instant answers now!

a 2 + 2 a 3 = 0

a = 3 , 1

Got questions? Get instant answers now!

x 2 + 9 x + 14 = 0

x = 7 , 2

Got questions? Get instant answers now!

b 2 + 12 b + 27 = 0

b = 9 , 3

Got questions? Get instant answers now!

x 2 13 x = 42

x = 6 , 7

Got questions? Get instant answers now!

a 3 = 8 a 2 15 a

Got questions? Get instant answers now!

6 a 2 + 13 a + 5 = 0

a = 5 3 , 1 2

Got questions? Get instant answers now!

12 a 2 + 15 a + 3 = 0

a = 1 4 , 1

Got questions? Get instant answers now!

12 a 2 + 24 a + 12 = 0

a = 1

Got questions? Get instant answers now!

2 x 2 = x + 15

x = 5 2 , 3

Got questions? Get instant answers now!

4 y 2 = 4 y 2

no solution

Got questions? Get instant answers now!

Exercises for review

( [link] ) Simplify ( x 4 y 3 ) 2 ( x y 2 ) 4 .

x 12 y 14

Got questions? Get instant answers now!

( [link] ) Write ( x 2 y 3 w 4 ) 2 so that only positive exponents appear.

Got questions? Get instant answers now!

( [link] ) Find the sum: x x 2 x 2 + 1 x 2 3 x + 2 .

x 2 + 1 ( x + 1 ) ( x 1 ) ( x 2 )

Got questions? Get instant answers now!

( [link] ) Simplify 1 a + 1 b 1 a 1 b .

Got questions? Get instant answers now!

( [link] ) Solve ( x + 4 ) ( 3 x + 1 ) = 0.

x = 4 , 1 3

Got questions? Get instant answers now!

Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask