Quadratic Equations Notes | EduRev

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: Quadratic Equations Notes | EduRev

 Page 1


Section 1.4     Quadratic Equations  33 
Larson/Hostetler  Algebra and Trigonometry, Sixth Edition  Student Success Organizer 
Copyright © Houghton Mifflin Company. All rights reserved. 
Section 1.4 Quadratic Equations 
 
Objective: In this lesson you learned how to solve quadratic equations. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
I.  Factoring  (Page 109) 
 
To use the Zero-Factor Property to solve a quadratic equation,               
. . .      aaaaa aaa aaaa aaaa aa aaa aaaaaaa aaaa aa a aaaaaaaaa 
aaaaaaaa aa aaa aaaaaaa aa aaa aaaaaa aaaaaaaa aaaa aaaa aaa 
aaaaaaaaa aa aaa aaaaaaaaa aaaaaaaa aa aaaaaaa aaaa aaaaaa 
aaaaaa aaaaa aa aaaaa 
 
Example 1: Solve 27 12
2
- = - x x by factoring. 
aaa aaaaaaaaa aaa a aaa aa 
 
 
II.  Extracting Square Roots  (Page 110) 
 
Solving an equation of the form d u =
2
 without going through 
the steps of factoring is called         aaaaaaaaaa aaaa aaaaa          . 
 
The equation d u =
2
, where d > 0, has exactly two solutions:        
u =          a a                  and u =         a a a                  . These 
solutions can also be written as u =         a a a                  . 
 
Example 2: Solve 45 ) 4 ( 5
2
= - x by extracting square roots. 
aaa aaaaaaaaa aaa a aaa aa 
 
 
 
 
Course Number 
 
Instructor 
 
Date 
 
Important Vocabulary  Define each term or concept. 
 
Quadratic equation  aa aaaaaaaa aa a aaaa aaa aa aaaaaaa aa aaa aaaaaaa aaaa                          
aa
a
 a aa a a a a aaaaa aa aa aaa a aaa aaaa aaaaaaa aaaa a a aa 
Quadratic formula  aaa aaaaaaaa a a aa a a a a
a
 a aaa a a aaaa aaa aaaaaaa aaaaaaaaa 
aa a aaaaaaaaa aaaaaaaa aa aaa aaaaaaa aaaa aa
a
 a aa a a a aa 
 
Discriminant  aaa aaaaaaaa aaaaa aaa aaaaaaa aaaaa a
a
 a aaaa aa aaa aaaaaaaaa 
aaaaaaaa 
 
What you should learn 
How to solve quadratic 
equations by factoring 
What you should learn 
How to solve quadratic 
equations by extracting 
square roots 
Page 2


Section 1.4     Quadratic Equations  33 
Larson/Hostetler  Algebra and Trigonometry, Sixth Edition  Student Success Organizer 
Copyright © Houghton Mifflin Company. All rights reserved. 
Section 1.4 Quadratic Equations 
 
Objective: In this lesson you learned how to solve quadratic equations. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
I.  Factoring  (Page 109) 
 
To use the Zero-Factor Property to solve a quadratic equation,               
. . .      aaaaa aaa aaaa aaaa aa aaa aaaaaaa aaaa aa a aaaaaaaaa 
aaaaaaaa aa aaa aaaaaaa aa aaa aaaaaa aaaaaaaa aaaa aaaa aaa 
aaaaaaaaa aa aaa aaaaaaaaa aaaaaaaa aa aaaaaaa aaaa aaaaaa 
aaaaaa aaaaa aa aaaaa 
 
Example 1: Solve 27 12
2
- = - x x by factoring. 
aaa aaaaaaaaa aaa a aaa aa 
 
 
II.  Extracting Square Roots  (Page 110) 
 
Solving an equation of the form d u =
2
 without going through 
the steps of factoring is called         aaaaaaaaaa aaaa aaaaa          . 
 
The equation d u =
2
, where d > 0, has exactly two solutions:        
u =          a a                  and u =         a a a                  . These 
solutions can also be written as u =         a a a                  . 
 
Example 2: Solve 45 ) 4 ( 5
2
= - x by extracting square roots. 
aaa aaaaaaaaa aaa a aaa aa 
 
 
 
 
Course Number 
 
Instructor 
 
Date 
 
Important Vocabulary  Define each term or concept. 
 
Quadratic equation  aa aaaaaaaa aa a aaaa aaa aa aaaaaaa aa aaa aaaaaaa aaaa                          
aa
a
 a aa a a a a aaaaa aa aa aaa a aaa aaaa aaaaaaa aaaa a a aa 
Quadratic formula  aaa aaaaaaaa a a aa a a a a
a
 a aaa a a aaaa aaa aaaaaaa aaaaaaaaa 
aa a aaaaaaaaa aaaaaaaa aa aaa aaaaaaa aaaa aa
a
 a aa a a a aa 
 
Discriminant  aaa aaaaaaaa aaaaa aaa aaaaaaa aaaaa a
a
 a aaaa aa aaa aaaaaaaaa 
aaaaaaaa 
 
What you should learn 
How to solve quadratic 
equations by factoring 
What you should learn 
How to solve quadratic 
equations by extracting 
square roots 
34  Chapter 1     Equations and Inequalities 
 Larson/Hostetler  Algebra and Trigonometry, Sixth Edition  Student Success Organizer 
  Copyright © Houghton Mifflin Company. All rights reserved. 
III.  Completing the Square  (Page 111) 
 
To complete the square for the expression bx x +
2
, add  
     aaaaa
a
     , which is the square of half the coefficient of x. 
When solving quadratic equations by completing the square, you 
must add this term to        aaaa aaaa               in order to maintain 
equality. 
The completing the square method can be used to solve a 
quadratic equation when . . .     aaa aaaaaaaa aa aaa aaaaaaaaaaa 
 
When completing the square to solve a quadratic equation, if the 
leading coefficient is not 1, . . .    aaa aaaa aaaaaa aaaa aaaa aa 
aaa aaaaaaaa aa aaa aaaaaaa aaaaaaaaaaa aaaaaa aaaaaaaaaa aaa 
aaaaaaa 
 
Example 3: Solve 0 8 10
2
= - + x x by completing the square. 
aaaaaaa aa aaa aaaaaaa aaaaaaa aaa aaaaaaaaa aaa       
a aaaaa aaa aaaaa 
 
 
IV.  The Quadratic Formula  (Pages 112-113) 
 
The verbal statement of the Quadratic Formula is . . .      
aaaaaaaaa aa aaaa aa aaaaa aaa aaaaaa aaaa aa a aaaaaaa aaaaa 
aaaa aaa aaaaaaa aa aaaa 
When using the Quadratic Formula, remember that before the 
formula can be applied, . . .     aaa aaaa aaaaa aaaaa aaa 
aaaaaaaaa aaaaaaaa aa aaaaaaa aaaa aa
a
 a aa aa a aa 
 
Example 4: For the quadratic equation 
2
2 3 16 x x - = - , find 
the values of a, b, and c to be substituted into the 
Quadratic Formula. 
 a a aa a a a aa aaa a a aaa aa a a a aa a a aa aaa                 
a a a aa 
 
 
The discriminant of the quadratic expression c bx ax + +
2
 can 
be used to . . .     aaaaaaaaa aaa aaaaaa aa aaa aaaaaaaaa aa a 
aaaaaaaaa aaaaaaaaa 
What you should learn 
How to solve quadratic 
equations by completing 
the square 
What you should learn 
How to use the Quadratic 
Formula to solve 
quadratic equations 
Page 3


Section 1.4     Quadratic Equations  33 
Larson/Hostetler  Algebra and Trigonometry, Sixth Edition  Student Success Organizer 
Copyright © Houghton Mifflin Company. All rights reserved. 
Section 1.4 Quadratic Equations 
 
Objective: In this lesson you learned how to solve quadratic equations. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
I.  Factoring  (Page 109) 
 
To use the Zero-Factor Property to solve a quadratic equation,               
. . .      aaaaa aaa aaaa aaaa aa aaa aaaaaaa aaaa aa a aaaaaaaaa 
aaaaaaaa aa aaa aaaaaaa aa aaa aaaaaa aaaaaaaa aaaa aaaa aaa 
aaaaaaaaa aa aaa aaaaaaaaa aaaaaaaa aa aaaaaaa aaaa aaaaaa 
aaaaaa aaaaa aa aaaaa 
 
Example 1: Solve 27 12
2
- = - x x by factoring. 
aaa aaaaaaaaa aaa a aaa aa 
 
 
II.  Extracting Square Roots  (Page 110) 
 
Solving an equation of the form d u =
2
 without going through 
the steps of factoring is called         aaaaaaaaaa aaaa aaaaa          . 
 
The equation d u =
2
, where d > 0, has exactly two solutions:        
u =          a a                  and u =         a a a                  . These 
solutions can also be written as u =         a a a                  . 
 
Example 2: Solve 45 ) 4 ( 5
2
= - x by extracting square roots. 
aaa aaaaaaaaa aaa a aaa aa 
 
 
 
 
Course Number 
 
Instructor 
 
Date 
 
Important Vocabulary  Define each term or concept. 
 
Quadratic equation  aa aaaaaaaa aa a aaaa aaa aa aaaaaaa aa aaa aaaaaaa aaaa                          
aa
a
 a aa a a a a aaaaa aa aa aaa a aaa aaaa aaaaaaa aaaa a a aa 
Quadratic formula  aaa aaaaaaaa a a aa a a a a
a
 a aaa a a aaaa aaa aaaaaaa aaaaaaaaa 
aa a aaaaaaaaa aaaaaaaa aa aaa aaaaaaa aaaa aa
a
 a aa a a a aa 
 
Discriminant  aaa aaaaaaaa aaaaa aaa aaaaaaa aaaaa a
a
 a aaaa aa aaa aaaaaaaaa 
aaaaaaaa 
 
What you should learn 
How to solve quadratic 
equations by factoring 
What you should learn 
How to solve quadratic 
equations by extracting 
square roots 
34  Chapter 1     Equations and Inequalities 
 Larson/Hostetler  Algebra and Trigonometry, Sixth Edition  Student Success Organizer 
  Copyright © Houghton Mifflin Company. All rights reserved. 
III.  Completing the Square  (Page 111) 
 
To complete the square for the expression bx x +
2
, add  
     aaaaa
a
     , which is the square of half the coefficient of x. 
When solving quadratic equations by completing the square, you 
must add this term to        aaaa aaaa               in order to maintain 
equality. 
The completing the square method can be used to solve a 
quadratic equation when . . .     aaa aaaaaaaa aa aaa aaaaaaaaaaa 
 
When completing the square to solve a quadratic equation, if the 
leading coefficient is not 1, . . .    aaa aaaa aaaaaa aaaa aaaa aa 
aaa aaaaaaaa aa aaa aaaaaaa aaaaaaaaaaa aaaaaa aaaaaaaaaa aaa 
aaaaaaa 
 
Example 3: Solve 0 8 10
2
= - + x x by completing the square. 
aaaaaaa aa aaa aaaaaaa aaaaaaa aaa aaaaaaaaa aaa       
a aaaaa aaa aaaaa 
 
 
IV.  The Quadratic Formula  (Pages 112-113) 
 
The verbal statement of the Quadratic Formula is . . .      
aaaaaaaaa aa aaaa aa aaaaa aaa aaaaaa aaaa aa a aaaaaaa aaaaa 
aaaa aaa aaaaaaa aa aaaa 
When using the Quadratic Formula, remember that before the 
formula can be applied, . . .     aaa aaaa aaaaa aaaaa aaa 
aaaaaaaaa aaaaaaaa aa aaaaaaa aaaa aa
a
 a aa aa a aa 
 
Example 4: For the quadratic equation 
2
2 3 16 x x - = - , find 
the values of a, b, and c to be substituted into the 
Quadratic Formula. 
 a a aa a a a aa aaa a a aaa aa a a a aa a a aa aaa                 
a a a aa 
 
 
The discriminant of the quadratic expression c bx ax + +
2
 can 
be used to . . .     aaaaaaaaa aaa aaaaaa aa aaa aaaaaaaaa aa a 
aaaaaaaaa aaaaaaaaa 
What you should learn 
How to solve quadratic 
equations by completing 
the square 
What you should learn 
How to use the Quadratic 
Formula to solve 
quadratic equations 
Section 1.4     Quadratic Equations  35 
Larson/Hostetler  Algebra and Trigonometry, Sixth Edition  Student Success Organizer 
Copyright © Houghton Mifflin Company. All rights reserved. 
If the discriminant ac b 4
2
- of the quadratic equation 
0
2
= + + c bx ax , 0 ? a , is: 
1) positive, then the quadratic equation . . .                          
aaa aaa aaaaaaaa aaaa aaaaaaaaa aaa aaa aaaaa aaa aaa                   
aaaaaaaaaaaaa 
2) zero, then the quadratic equation . . .                                           
aaa aaa aaaaaaaa aaaa aaaaaaaa aaa aaa aaaaa aaa aaa                                
aaaaaaaaaaaa 
3) negative, then the quadratic equation . . .                          
aaa aa aaaa aaaaaaaaa aaa aaa aaaaa aaa aa aaaaaaaaaaaa 
 
Example 5: Use the discriminant to find the number and type 
of solutions of the quadratic equation 
0 18 5 6
2
= + - x x . 
 aa aaaa aaaaaaaaa 
 
 
V.  Applications of Quadratic Equations  (Pages 114-117) 
 
Describe two real -life situations in which quadratic equations 
often occur. 
 
aaaaaaa aaaa aaaaa aaaaaaaa aaaaaaaaa aaaaaaa aaaaaaaa aaaaaaa 
aaaa aaaa aaa aaaaaaaa aaaaaaa aaaa aaa aaaaaaaaaa aa a aaaaa 
aaaaaaaaa 
 
 
The position equation giving the height of an object above the 
Earth’s surface is         a a a aaa
a
 aa
a
a a a
a
            , where . . .      
a aaaaaaaaaa aaa aaaaaa aa aaa aaaaaa aa aaaaa a
a
 aaaaaaaaaa 
aaa aaaaaaa aaaaaaaa aa aaa aaaaaa aa aaaa aaa aaaaaaa a
a
 
aaaaaaaaaa aaa aaaaaaa aaaaaa aa aaa aaaaaa aa aaaaa aaa a 
aaaaaaaaaa aaaa aa aaaaaaaa 
 
 
 
Homework Assignment 
Page(s) 
 
Exercises 
What you should learn 
How to use quadratic 
equations to model and 
solve real -life problems 
Page 4


Section 1.4     Quadratic Equations  33 
Larson/Hostetler  Algebra and Trigonometry, Sixth Edition  Student Success Organizer 
Copyright © Houghton Mifflin Company. All rights reserved. 
Section 1.4 Quadratic Equations 
 
Objective: In this lesson you learned how to solve quadratic equations. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
I.  Factoring  (Page 109) 
 
To use the Zero-Factor Property to solve a quadratic equation,               
. . .      aaaaa aaa aaaa aaaa aa aaa aaaaaaa aaaa aa a aaaaaaaaa 
aaaaaaaa aa aaa aaaaaaa aa aaa aaaaaa aaaaaaaa aaaa aaaa aaa 
aaaaaaaaa aa aaa aaaaaaaaa aaaaaaaa aa aaaaaaa aaaa aaaaaa 
aaaaaa aaaaa aa aaaaa 
 
Example 1: Solve 27 12
2
- = - x x by factoring. 
aaa aaaaaaaaa aaa a aaa aa 
 
 
II.  Extracting Square Roots  (Page 110) 
 
Solving an equation of the form d u =
2
 without going through 
the steps of factoring is called         aaaaaaaaaa aaaa aaaaa          . 
 
The equation d u =
2
, where d > 0, has exactly two solutions:        
u =          a a                  and u =         a a a                  . These 
solutions can also be written as u =         a a a                  . 
 
Example 2: Solve 45 ) 4 ( 5
2
= - x by extracting square roots. 
aaa aaaaaaaaa aaa a aaa aa 
 
 
 
 
Course Number 
 
Instructor 
 
Date 
 
Important Vocabulary  Define each term or concept. 
 
Quadratic equation  aa aaaaaaaa aa a aaaa aaa aa aaaaaaa aa aaa aaaaaaa aaaa                          
aa
a
 a aa a a a a aaaaa aa aa aaa a aaa aaaa aaaaaaa aaaa a a aa 
Quadratic formula  aaa aaaaaaaa a a aa a a a a
a
 a aaa a a aaaa aaa aaaaaaa aaaaaaaaa 
aa a aaaaaaaaa aaaaaaaa aa aaa aaaaaaa aaaa aa
a
 a aa a a a aa 
 
Discriminant  aaa aaaaaaaa aaaaa aaa aaaaaaa aaaaa a
a
 a aaaa aa aaa aaaaaaaaa 
aaaaaaaa 
 
What you should learn 
How to solve quadratic 
equations by factoring 
What you should learn 
How to solve quadratic 
equations by extracting 
square roots 
34  Chapter 1     Equations and Inequalities 
 Larson/Hostetler  Algebra and Trigonometry, Sixth Edition  Student Success Organizer 
  Copyright © Houghton Mifflin Company. All rights reserved. 
III.  Completing the Square  (Page 111) 
 
To complete the square for the expression bx x +
2
, add  
     aaaaa
a
     , which is the square of half the coefficient of x. 
When solving quadratic equations by completing the square, you 
must add this term to        aaaa aaaa               in order to maintain 
equality. 
The completing the square method can be used to solve a 
quadratic equation when . . .     aaa aaaaaaaa aa aaa aaaaaaaaaaa 
 
When completing the square to solve a quadratic equation, if the 
leading coefficient is not 1, . . .    aaa aaaa aaaaaa aaaa aaaa aa 
aaa aaaaaaaa aa aaa aaaaaaa aaaaaaaaaaa aaaaaa aaaaaaaaaa aaa 
aaaaaaa 
 
Example 3: Solve 0 8 10
2
= - + x x by completing the square. 
aaaaaaa aa aaa aaaaaaa aaaaaaa aaa aaaaaaaaa aaa       
a aaaaa aaa aaaaa 
 
 
IV.  The Quadratic Formula  (Pages 112-113) 
 
The verbal statement of the Quadratic Formula is . . .      
aaaaaaaaa aa aaaa aa aaaaa aaa aaaaaa aaaa aa a aaaaaaa aaaaa 
aaaa aaa aaaaaaa aa aaaa 
When using the Quadratic Formula, remember that before the 
formula can be applied, . . .     aaa aaaa aaaaa aaaaa aaa 
aaaaaaaaa aaaaaaaa aa aaaaaaa aaaa aa
a
 a aa aa a aa 
 
Example 4: For the quadratic equation 
2
2 3 16 x x - = - , find 
the values of a, b, and c to be substituted into the 
Quadratic Formula. 
 a a aa a a a aa aaa a a aaa aa a a a aa a a aa aaa                 
a a a aa 
 
 
The discriminant of the quadratic expression c bx ax + +
2
 can 
be used to . . .     aaaaaaaaa aaa aaaaaa aa aaa aaaaaaaaa aa a 
aaaaaaaaa aaaaaaaaa 
What you should learn 
How to solve quadratic 
equations by completing 
the square 
What you should learn 
How to use the Quadratic 
Formula to solve 
quadratic equations 
Section 1.4     Quadratic Equations  35 
Larson/Hostetler  Algebra and Trigonometry, Sixth Edition  Student Success Organizer 
Copyright © Houghton Mifflin Company. All rights reserved. 
If the discriminant ac b 4
2
- of the quadratic equation 
0
2
= + + c bx ax , 0 ? a , is: 
1) positive, then the quadratic equation . . .                          
aaa aaa aaaaaaaa aaaa aaaaaaaaa aaa aaa aaaaa aaa aaa                   
aaaaaaaaaaaaa 
2) zero, then the quadratic equation . . .                                           
aaa aaa aaaaaaaa aaaa aaaaaaaa aaa aaa aaaaa aaa aaa                                
aaaaaaaaaaaa 
3) negative, then the quadratic equation . . .                          
aaa aa aaaa aaaaaaaaa aaa aaa aaaaa aaa aa aaaaaaaaaaaa 
 
Example 5: Use the discriminant to find the number and type 
of solutions of the quadratic equation 
0 18 5 6
2
= + - x x . 
 aa aaaa aaaaaaaaa 
 
 
V.  Applications of Quadratic Equations  (Pages 114-117) 
 
Describe two real -life situations in which quadratic equations 
often occur. 
 
aaaaaaa aaaa aaaaa aaaaaaaa aaaaaaaaa aaaaaaa aaaaaaaa aaaaaaa 
aaaa aaaa aaa aaaaaaaa aaaaaaa aaaa aaa aaaaaaaaaa aa a aaaaa 
aaaaaaaaa 
 
 
The position equation giving the height of an object above the 
Earth’s surface is         a a a aaa
a
 aa
a
a a a
a
            , where . . .      
a aaaaaaaaaa aaa aaaaaa aa aaa aaaaaa aa aaaaa a
a
 aaaaaaaaaa 
aaa aaaaaaa aaaaaaaa aa aaa aaaaaa aa aaaa aaa aaaaaaa a
a
 
aaaaaaaaaa aaa aaaaaaa aaaaaa aa aaa aaaaaa aa aaaaa aaa a 
aaaaaaaaaa aaaa aa aaaaaaaa 
 
 
 
Homework Assignment 
Page(s) 
 
Exercises 
What you should learn 
How to use quadratic 
equations to model and 
solve real -life problems 
36  Chapter 1     Equations and Inequalities 
 Larson/Hostetler  Algebra and Trigonometry, Sixth Edition  Student Success Organizer 
  Copyright © Houghton Mifflin Company. All rights reserved. 
 
 
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