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# Quadratic Equations Notes | EduRev

## : Quadratic Equations Notes | EduRev

Page 1

Section 1.4     Quadratic Equations  33
Larson/Hostetler  Algebra and Trigonometry, Sixth Edition  Student Success Organizer
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
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
III.  Completing the Square  (Page 111)

To complete the square for the expression bx x +
2
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
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
Page 3

Section 1.4     Quadratic Equations  33
Larson/Hostetler  Algebra and Trigonometry, Sixth Edition  Student Success Organizer
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
III.  Completing the Square  (Page 111)

To complete the square for the expression bx x +
2
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
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
Section 1.4     Quadratic Equations  35
Larson/Hostetler  Algebra and Trigonometry, Sixth Edition  Student Success Organizer
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
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
III.  Completing the Square  (Page 111)

To complete the square for the expression bx x +
2
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
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
Section 1.4     Quadratic Equations  35
Larson/Hostetler  Algebra and Trigonometry, Sixth Edition  Student Success Organizer
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