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

• Solve the following:

• 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

• Solve the following:

• 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

• Solve the following:

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

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

02452 xx

_)_)(( xx

Factoring

• Solve the following:

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

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

02452 xx

0)8)(3( xx

Factoring

• Solve the following:

• 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

• Solve the following: 024102 xx

Factoring

• Solve the following:

• 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

• Solve the following:

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

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

024102 xx

_)_)(( xx

Factoring

• Solve the following:

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

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

024102 xx

0)4)(6( xx

Factoring

• Solve the following:

• Solve:

or

024102 xx

0)4)(6( xx

6

06

x

x

4

04

x

x

You Try

• Solve the following: 036132 xx

You Try

• Solve the following:

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

• We need factors of 36 that add to get 13.

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:

• First, set the equation equal to zero.

• This is how we start:

3792 2 xx

0492 2 xx

_)_)(2( xx

Factoring

• Solve:

• First, set the equation equal to zero.

• 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:

• First, set the equation equal to zero.

• 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.

Completing the Square

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

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

Completing the Square

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

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

Completing the Square

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

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.

Completing the Square

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

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.

Completing the Square

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

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.

Completing the Square

• Solve: 0822 xx

Completing the Square

• Solve:

• Add eight to both sides.

0822 xx

822 xx

Completing the Square

• Solve:

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

0822 xx

18122 xx

822 xx

Completing the Square

• Solve:

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

• Solve.

0822 xx

9)1( 2 x

822 xx

18122 xx

31 x

Completing the Square

• Solve:

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

• Solve.

0822 xx

9)1( 2 x

822 xx

18122 xx

31 x

2

31

x

x

4

31

x

x

Completing the Square

• You Try:0382 xx

Completing the Square

• Solve:0382 xx

382 xx

Completing the Square

• Solve:0382 xx

382 xx

163)4(8 22 xx

Completing the Square

• Solve:0382 xx

382 xx

1631682 xx

13)4( 2 x

134 x 134x

Completing the Square

• You Try:01642 xx

Completing the Square

• Solve:01642 xx

1642 xx

416442 xx

20)2( 2 x

Completing the Square

• Solve:01642 xx

1642 xx

416442 xx

20)2( 2 x

202 x 522x

Completing the Square

• Solve: 0382 2 xx

382 2 xx

2/342 xx

Completing the Square

• Solve: 0382 2 xx

382 2 xx

2/342 xx

42/3442 xx

2/5)2( 2 x

Completing the Square

• Solve: 0382 2 xx

382 2 xx

2/342 xx

42/3442 xx

2/5)2( 2 x

2/52 x

Completing the Square

• Solve: 0382 2 xx

382 2 xx

2/342 xx

42/3442 xx

2/5)2( 2 x

2/52 x

2/52x

Completing the Square

• 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

• Rationalize the following:2

5

2

5

Rationalize

• Rationalize the following:2

5

2

2

2

5

Rationalize

• Rationalize the following:2

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

The Quadratic Formula

• Solve using the quadratic formula. 932 xx

The Quadratic Formula

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

932 xx

0932 xx

The Quadratic Formula

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

932 xx

0932 xx

The Quadratic Formula

• 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

The Quadratic Formula

• 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

2

3693)1(2

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

The Quadratic Formula

• 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

2

3693)1(2

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

2

533

2

453

The Quadratic Formula

• Solve using the quadratic formula. 462 xx

The Quadratic Formula

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

462 xx

0462 xx

The Quadratic Formula

• 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

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

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

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.

• 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

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

1809.12519.17163.1 2 tt

01.5419.17163.1 2 tt

Ex. 9

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

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

• 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?

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

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?

• 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