<< 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: recognize the standard form of a quadratic equation, understand the derivation of the quadratic formula, solve quadratic equations using the quadratic formula.

Overview

  • Standard Form Of A Quadratic Equation
  • The Quadratic Formula
  • Derivation Of The Quadratic Formula

Standard form of a quadratic equation

We have observed that a quadratic equation is an equation of the form

a x 2 + b x + c = 0 , a 0

where

a is the coefficient of the quadratic term,
b is the coefficient of the linear term, and
c is the constant term.

Standard form

The equation a x 2 + b x + c = 0 is the standard form of a quadratic equation.

Sample set a

Determine the values of a , b , and c .

In the equation 3 x 2 + 5 x + 2 = 0 ,

a = 3 b = 5 c = 2

Got questions? Get instant answers now!

In the equation 12 x 2 2 x 1 = 0 ,

a = 12 b = 2 c = 1

Got questions? Get instant answers now!

In the equation 2 y 2 + 3 = 0 ,

a = 2 b = 0 Because the equation could be written 2 y 2 + 0 y + 3 = 0 c = 3

Got questions? Get instant answers now!

In the equation 8 y 2 + 11 y = 0 ,

a = 8 b = 11 c = 0 Since  8y 2 + 11 y + 0 = 0.

Got questions? Get instant answers now!

In the equation z 2 = z + 8 ,

a = 1 b = 1 c = 8 When we write the equation in standard form, we get  z 2 z 8 = 0

Got questions? Get instant answers now!

Practice set a

Determine the values of a , b , and c in the following quadratic equations.

4 x 2 3 x + 5 = 0

a = 4 b = 3 c = 5

Got questions? Get instant answers now!

3 y 2 2 y + 9 = 0

a = 3 b = 2 c = 9

Got questions? Get instant answers now!

x 2 5 x 1 = 0

a = 1 b = 5 c = 1

Got questions? Get instant answers now!

z 2 4 = 0

a = 1 b = 0 c = 4

Got questions? Get instant answers now!

x 2 2 x = 0

a = 1 b = 2 c = 0

Got questions? Get instant answers now!

y 2 = 5 y 6

a = 1 b = 5 c = 6

Got questions? Get instant answers now!

2 x 2 4 x = 1

a = 2 b = 4 c = 1

Got questions? Get instant answers now!

5 x 3 = 3 x 2

a = 3 b = 5 c = 3

Got questions? Get instant answers now!

2 x 11 3 x 2 = 0

a = 3 b = 2 c = 11

Got questions? Get instant answers now!

y 2 = 0

a = 1 b = 0 c = 0

Got questions? Get instant answers now!

The solutions to all quadratic equations depend only and completely on the values a , b , and c .

The quadratic formula

When a quadratic equation is written in standard form so that the values a , b , and c are readily determined, the equation can be solved using the quadratic formula . The values that satisfy the equation are found by substituting the values a , b , and c into the formula

Quadratic formula

x = b ± b 2 4 a c 2 a

Keep in mind that the plus or minus symbol, ± , is just a shorthand way of denoting the two possibilities:

x = b + b 2 4 a c 2 a and x = b b 2 4 a c 2 a

The quadratic formula can be derived by using the method of completing the square.

Derivation of the quadratic formula

Solve a x 2 + b x + c = 0 for x by completing the square.

Subtract c from both sides.

a x 2 + b x = c

Got questions? Get instant answers now!

Divide both sides by a , the coefficient of x 2 .

x 2 + b a x = c a

Got questions? Get instant answers now!

Now we have the proper form to complete the square. Take one half the coefficient of x , square it, and add the result to both sides of the equation found in step 2.

(a) 1 2 · b a = b 2 a is one half the coefficient of x .

(b) ( b 2 a ) 2 is the square of one half the coefficient of x .

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

Got questions? Get instant answers now!

The left side of the equation is now a perfect square trinomial and can be factored. This gives us

( x + b 2 a ) 2 = c a + b 2 4 a 2

Got questions? Get instant answers now!

Add the two fractions on the right side of the equation. The LCD = 4 a 2 .

( x + b 2 a ) 2 = 4 a c 4 a 2 + b 2 4 a 2 ( x + b 2 a ) 2 = 4 a c + b 2 4 a 2 ( x + b 2 a ) 2 = b 2 4 a c 4 a 2

Got questions? Get instant answers now!

Solve for x using the method of extraction of roots.

x + b 2 a = ± b 2 4 a c 4 a 2 x + b 2 a = ± b 2 4 a c 4 a 2 4 a 2 = | 2 a | = 2 | a | = ± 2 a x + b 2 a = ± b 2 4 a c 2 a x = b 2 a ± b 2 4 a c 2 a Add these two fractions . x = b ± b 2 4 a c 2 a

Got questions? Get instant answers now!

Sample set b

Solve each of the following quadratic equations using the quadratic formula.

3 x 2 + 5 x + 2 = 0.

  1. Identify a , b , and c .

    a = 3 , b = 5 , and c = 2
  2. Write the quadratic formula.

    x = b ± b 2 4 a c 2 a
  3. Substitute.

    x = 5 ± ( 5 ) 2 4 ( 3 ) ( 2 ) 2 ( 3 ) = 5 ± 25 24 6 = 5 ± 1 6 = 5 ± 1 6 5 + 1 = 4  and  5 1 = 6 = 4 6 , 6 6 x = 2 3 , 1
    N o t e : Since these roots are rational numbers, this equation could have been solved by factoring .

Got questions? Get instant answers now!

12 x 2 2 x 1 = 0.

  1. Identify a , b , and c .

    a = 12 , b = 2 , and c = 1
  2. Write the quadratic formula.

    x = b ± b 2 4 a c 2 a
  3. Substitute.

    x = ( 2 ) ± ( 2 ) 2 4 ( 12 ) ( 1 ) 2 ( 12 ) = 2 ± 4 + 48 24 Simplify . = 2 ± 52 24 Simplify . = 2 ± 4 · 13 24 Simplify . = 2 ± 2 13 24 Reduce .  Factor 2 from the terms of the numerator . = 2 ( 1 ± 13 ) 24 x = 1 ± 13 12

Got questions? Get instant answers now!

2 y 2 + 3 = 0

  1. Identify a , b , and c .

    a = 2 , b = 0 , and c = 3
  2. Write the quadratic formula.

    x = b ± b 2 4 a c 2 a
  3. Substitute.

    x = 0 ± 0 2 4 ( 2 ) ( 3 ) 2 ( 2 ) x = 0 ± 24 4

    This equation has no real number solution since we have obtained a negative number under the radical sign.

Got questions? Get instant answers now!

8 x 2 + 11 x = 0

  1. Identify a , b , and c .

    a = 8 , b = 11 , and c = 0
  2. Write the quadratic formula.

    x = b ± b 2 4 a c 2 a
  3. Substitute.

    x = 11 ± 11 2 4 ( 8 ) ( 0 ) 2 ( 8 ) = 11 ± 121 0 16 Simplify . = 11 ± 121 16 Simplify . = 11 ± 11 16 x = 0 , 11 8

Got questions? Get instant answers now!

( 3 x + 1 ) ( x 4 ) = x 2 + x 2

  1. Write the equation in standard form.

    3 x 2 11 x 4 = x 2 + x 2 2 x 2 12 x 2 = 0 x 2 6 x 1 = 0
  2. Identify a , b , and c .

    a = 1 , b = 6 , and c = 1
  3. Write the quadratic formula.

    x = b ± b 2 4 a c 2 a
  4. Substitute.

    x = ( 6 ) ± ( 6 ) 2 4 ( 1 ) ( 1 ) 2 ( 1 ) = 6 ± 36 + 4 2 = 6 ± 40 2 = 6 ± 4 · 10 2 = 6 ± 2 10 2 = 2 ( 3 ± 10 ) 2 x = 3 ± 10

Got questions? Get instant answers now!

Practice set b

Solve each of the following quadratic equations using the quadratic formula.

2 x 2 + 3 x 7 = 0

x = 3 ± 65 4

Got questions? Get instant answers now!

5 a 2 2 a 1 = 0

a = 1 ± 6 5

Got questions? Get instant answers now!

6 y 2 + 5 = 0

no real number solution

Got questions? Get instant answers now!

3 m 2 + 2 m = 0

m = 0 , 2 3

Got questions? Get instant answers now!

Exercises

For the following problems, solve the equations using the quadratic formula.

x 2 2 x 3 = 0

x = 3 , 1

Got questions? Get instant answers now!

y 2 5 y + 4 = 0

y = 1 , 4

Got questions? Get instant answers now!

a 2 + 12 a + 20 = 0

a = 2 , 10

Got questions? Get instant answers now!

b 2 + 4 b + 4 = 0

b = 2

Got questions? Get instant answers now!

2 x 2 5 x 3 = 0

x = 3 , 1 2

Got questions? Get instant answers now!

4 x 2 2 x 1 = 0

x = 1 ± 5 4

Got questions? Get instant answers now!

5 a 2 2 a 3 = 0

a = 1 , 3 5

Got questions? Get instant answers now!

x 2 5 x 4 = 0

x = 5 ± 41 2

Got questions? Get instant answers now!

( x + 2 ) ( x 1 ) = 1

Got questions? Get instant answers now!

( a + 4 ) ( a 5 ) = 2

a = 1 ± 89 2

Got questions? Get instant answers now!

( x 3 ) ( x + 3 ) = 7

Got questions? Get instant answers now!

( b 4 ) ( b + 4 ) = 9

b = ± 5

Got questions? Get instant answers now!

y 2 = 5 y + 4

y = 5 ± 41 2

Got questions? Get instant answers now!

x 2 = 2 x 1

x = 1

Got questions? Get instant answers now!

a 2 + 3 a 4 = 0

a = 4 , 1

Got questions? Get instant answers now!

b 2 + 3 b = 2

b = 1 , 2

Got questions? Get instant answers now!

x 2 + 6 x + 8 = x 2

Got questions? Get instant answers now!

x 2 + 4 x = 2 x 5

No real number solution.

Got questions? Get instant answers now!

6 b 2 + 5 b 4 = b 2 + b + 1

Got questions? Get instant answers now!

4 a 2 + 7 a 2 = 2 a + a

2 ± 6 2

Got questions? Get instant answers now!

( 2 x + 5 ) ( x 4 ) = x 2 x + 2

Got questions? Get instant answers now!

( x 4 ) 2 = 3

x = 4 ± 3

Got questions? Get instant answers now!

( b 6 ) 2 = 8

b = 6 ± 2 2

Got questions? Get instant answers now!

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

x = 1 ± 34 3

Got questions? Get instant answers now!

2 ( y 2 3 ) = 3 ( y 1 )

Got questions? Get instant answers now!

4 ( a 2 + 2 ) + 3 = 5

No real number solution.

Got questions? Get instant answers now!

( x 2 + 3 x 1 ) = 2

Got questions? Get instant answers now!

Exercises for review

( [link] ) Simplify ( x 8 y 7 z 5 x 4 y 6 z 2 ) 2 .

x 8 y 2 z 6

Got questions? Get instant answers now!

( [link] ) Write 4 a 6 b 2 c 3 a 5 b 3 so that only positive exponents appear.

Got questions? Get instant answers now!

( [link] ) Find the product: ( 2 y + 7 ) ( 3 y 1 ) .

6 y 2 + 19 y 7

Got questions? Get instant answers now!

( [link] ) Solve x 2 4 x 12 = 0 by completing the square.

x = 2 , 6

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