quadratic equations and applications objective: to use different methods to solve quadratic...

Quadratic Equations and Applications

Objective: To use different methods to solve quadratic

equations.

Factoring

• Factoring is an effective way to solve quadratic equations.

Factoring

• Solve the following: 02452 xx

Factoring

• Solve the following:

• This is the easiest type because you always start this way.

02452 xx

__)__)(( xx

Factoring


• This is the easiest type because you always start this way. Since the last term is negative, the signs must be opposite.

02452 xx

_)_)(( xx

Factoring


• Since the middle term is – 5x, that means that the outer and inner terms are different by 5, and the larger term is negative.

02452 xx

_)_)(( xx

Factoring



• We now need factors of 24 that are different by 5.

02452 xx

_)_)(( xx

Factoring



• We now need factors of 24 that are different by 5.

02452 xx

0)8)(3( xx

Factoring


• We know that if two things multiplied together equal zero, one of them must be zero.

or

02452 xx

0)8)(3( xx

3

03

x

x8

08

x

x

Factoring


Factoring


• Since the last term is positive, both factors are the same sign, and since the middle term is positive, they are both positive.

024102 xx

_)_)(( xx

Factoring



• We need factors of 24 that add to get 10.

024102 xx

_)_)(( xx

Factoring




024102 xx

0)4)(6( xx

Factoring


• Solve:

or

024102 xx

0)4)(6( xx

6

06

x

x

4

04

x

x

You Try


You Try


• Since the last term is positive, both factors are the same sign, and now since the middle term is negative, they are both negative.


036132 xx

0)4)(9( xx

9

09

x

x

4

04

x

x

Factoring

• Solve: 3792 2 xx

Factoring

• Solve:

• First, set the equation equal to zero.

3792 2 xx

0492 2 xx

Factoring

• Solve:


• This is how we start:

3792 2 xx

0492 2 xx

_)_)(2( xx

Factoring

• Solve:


• This is how we start:• We need factors of 4 that will make the outer and

inner terms add to get 9.

3792 2 xx

0492 2 xx

_)_)(2( xx

Factoring

• Solve:


• This is how we start:• We need factors of 4 that will make the outer and

inner terms add to get 9.

3792 2 xx

0492 2 xx

_)_)(2( xx

0)4)(12( xx 2/1

012

x

x

4

04

x

x

Factoring

• Solve: 036 2 xx

Factoring

• Solve:

• With only two terms, we look for a greatest common factor.

or

036 2 xx

0)12(3 xx

0

03

x

x

21

012

x

x

Square Roots

• Solve: 124 2 x

Square Roots

• Solve:

• First, isolate the variable and get the x2 alone.

124 2 x

32 x

Square Roots

• Solve:

• First, isolate the variable and get the x2 alone.

• Square root both sides.

124 2 x

32 x

3x

Square Roots

• Solve: 7)3( 2 x

Square Roots

• Solve:

• The x is by itself as much as it can be, so square root both sides.

7)3( 2 x

73 x

Square Roots

• Solve:

• The x is by itself as much as it can be, so square root both sides.

• Add 3 to both sides to solve for x.

7)3( 2 x

73 x

73x

Square Roots

• You Try: 16)6( 2 x

Square Roots

• You Try: 16)6( 2 x

46 x

1046

246

46

x

Class Work

• Page 120

• 8-14 even

• 22, 24, 26, 28

Homework

• Page 120

• 7-15 odd

• 21-31 odd

Completing the Square

• When completing the square, we follow the same technique each time.



1. Variables on one side, constants on the other.



1. Variables on one side, constants on the other.2. Make sure the x2 is a 1x2.



1. Variables on one side, constants on the other.2. Make sure the x2 is a 1x2.3. Take half of the middle term and square it.



1. Variables on one side, constants on the other.2. Make sure the x2 is a 1x2.3. Take half of the middle term and square it.4. Add that number to both sides.



1. Variables on one side, constants on the other.2. Make sure the x2 is a 1x2.3. Take half of the middle term and square it.4. Add that number to both sides.5. Solve.


• Solve: 0822 xx


• Solve:

• Add eight to both sides.

0822 xx

822 xx


• Solve:

• Add eight to both sides.• Take half of 2, square it, and add it to both sides.

0822 xx

18122 xx

822 xx


• Solve:


• Solve.

0822 xx

9)1( 2 x

822 xx

18122 xx

31 x


• Solve:


• Solve.

0822 xx

9)1( 2 x

822 xx

18122 xx

31 x

2

31

x

x

4

31

x

x


• You Try:0382 xx


• Solve:0382 xx

382 xx


• Solve:0382 xx

382 xx

163)4(8 22 xx


• Solve:0382 xx

382 xx

1631682 xx

13)4( 2 x

134 x 134x


• You Try:01642 xx


• Solve:01642 xx

1642 xx

416442 xx

20)2( 2 x


• Solve:01642 xx

1642 xx

416442 xx

20)2( 2 x

202 x 522x


• Solve: 0382 2 xx

382 2 xx

2/342 xx


• Solve: 0382 2 xx

382 2 xx

2/342 xx

42/3442 xx

2/5)2( 2 x


• Solve: 0382 2 xx

382 2 xx

2/342 xx

42/3442 xx

2/5)2( 2 x

2/52 x


• Solve: 0382 2 xx

382 2 xx

2/342 xx

42/3442 xx

2/5)2( 2 x

2/52 x

2/52x


• Solve: 0382 2 xx

382 2 xx

2/342 xx

42/3442 xx

2/5)2( 2 x

2/52 x

2/52x

2

102x

Rationalize

• Rationalize the following:2

5

Rationalize


5

2

5

Rationalize


5

2

2

2

5

Rationalize


5

2

10

2

2

2

5

Rationalize

• You Try.

7

3

11

5

Rationalize

• You Try.

7

3

11

5

7

21

7

7

7

3

11

55

11

11

11

5

The Quadratic Formula

• Given a quadratic equation of the form:

• We can solve it using the quadratic formula:

0,02 acbxax

a

acbb

2

42


• Solve using the quadratic formula. 932 xx


• Solve using the quadratic formula.• Set the equation equal to zero.

932 xx

0932 xx


• Solve using the quadratic formula.• Set the equation equal to zero.• a=1, b=3, c=-9

932 xx

0932 xx


• Solve using the quadratic formula.• Set the equation equal to zero.• a=1, b=3, c=-9• Use the quadratic equation.

932 xx

0932 xx

)1(2)9)(1(4)3(3 2



932 xx

0932 xx

2

3693)1(2

)9)(1(4)3(3 2



932 xx

0932 xx

2

3693)1(2

)9)(1(4)3(3 2

2

533

2

453


• Solve using the quadratic formula. 462 xx


• Solve using the quadratic formula.• Set the equation equal to zero.• a=1, b=-6, c=-4

462 xx

0462 xx


• Solve using the quadratic formula.• Set the equation equal to zero.• a=1, b=-6, c=-4• Use the quadratic equation.

462 xx

0462 xx

2

16366)1(2

)4)(1(4)6(6 2

1332

1326

2

526

Class Work

• Page 120

• 35, 37, 38

Class Work

• Page 120

• 35, 37, 38

• 68, 72, 74, 76

Example 7

• A bedroom is 3 feet longer than it is wide and has an area of 154 square feet. Find the dimensions of the room.

Example 7


• The width will be x• The length will be x + 3

Example 7


• The width will be x• The length will be x + 3• Since the area is length x width, the equation is:

x(x+3) = 154

Ex. 7

• We need to solve the equation to find x.

154)3( xx

015432 xx

Ex. 7

• We need to solve the equation to find x.

154)3( xx

015432 xx

0)14)(11( xx

11,011 xx 14,014 xx

Ex. 7

• Since x is the width of a room, it can’t be negative. The only answer we will use is x = 11.

• This means the width is 11 and the length is 14.

Example 9

• From 2000 to 2007, the estimated number of Internet users I (in millions) in the United States can be modeled by the quadratic equation

• where t represents the year, with t = 0 being 2000. In what year will the number of Internet users surpass 180 million?

,9.12519.17163.1 2 ttI 70 t

Ex. 9

• We will use our calculator to solve the following problem.

1809.12519.17163.1 2 tt

Ex. 9


1809.12519.17163.1 2 tt

01.5419.17163.1 2 tt

Ex. 9


1809.12519.17163.1 2 tt

01.5419.17163.1 2 tt

2.105.4 ort

2004

Example 10

• An L-shaped sidewalk from the athletic center to the library on a college campus is shown. The length of one sidewalk forming the L is twice as long as the other. The length of the diagonal between the two buildings is 32 feet. How many feet does a person save by walking on the diagonal?

Example 10


• Let x = one path• Let 2x = the other path

Example 10


• Let x = one path• Let 2x = the other path• We will use the Pythagorean Theorem to solve.

Ex. 10

222 32)2( xx

Ex. 10

222 32)2( xx

10245 2 x

Ex. 10

222 32)2( xx

10245 2 x

8.2042 x

Ex. 10

222 32)2( xx

10245 2 x

8.2042 x

8.204x

Ex. 10

222 32)2( xx

10245 2 x

8.2042 x

8.204x

31.14x

Ex. 10

222 32)2( xx

10245 2 x

8.2042 x

8.204x

31.14x

9.42)31.14(3

32

xxx

Ex. 10

222 32)2( xx

10245 2 x

8.2042 x

8.204x

31.14x

9.42)31.14(3

32

xxx

feet9.10329.42

Homework

• Page 120

• 21-42 mult. of 3

• 67-77 odd

• 109,124

Lesson Quiz

• Solve by completing the square.1. 01162 xx

523

203

20)3(

91196

116

2

2

2

x

x

x

xx

xx

Lesson Quiz

• Solve by completing the square.2. 0542 xx

1,5

32

9)2(

4544

54

2

2

2

x

x

x

xx

xx

Lesson Quiz

• Solve by using the quadratic formula.3. 0232 2 xx

)2(2

)2)(2(4)3(3 2

21,2

4

253

Lesson Quiz

• Solve by using the quadratic formula4. 0243 2 xx

)3(2

)2)(3(4)4(4 2

3

102

6

1024

6

404

quadratic equations and applications objective: to use different methods to solve quadratic...

Documents

factoring factoring

middle term

larger term

square roots

inner terms

equal zero

equation equal

different methods