TEMPLATE 1 PPT.ppt

Many quadratic equations can not be solved by factoring. Other techniques are required to solve them.

8.1 – Solving Quadratic Equations

x2 = 20 5x2 + 55 = 0

Examples:

( x + 2)2 = 18 ( 3x – 1)2 = –4

x2 + 8x = 1 2x2 – 2x + 7 = 0

22 5 0x x 44 2 xx

If b is a real number and if a2 = b, then a = ±√¯‾.

20

8.1 – Solving Quadratic EquationsSquare Root Property

b

x2 = 20

x = ±√‾‾

x = ±√‾‾‾‾4·5

x = ± 2√‾ 5 –11

5x2 + 55 = 0

x = ±√‾‾‾

5x2 = –55

x2 = –11

x = ± i√‾‾‾11

If b is a real number and if a2 = b, then a = ±√¯‾.

18

8.1 – Solving Quadratic EquationsSquare Root Property

b

( x + 2)2 = 18

x + 2 = ±√‾‾

x + 2 = ±√‾‾‾‾9·2

x +2 = ± 3√‾ 2

x = –2 ± 3√‾ 2

–4

( 3x – 1)2 = –4

3x – 1 = ±√‾‾

3x – 1 = ± 2i

3x = 1 ± 2i

3

21 ix

ix3

2

3

1

Review:

8.1 – Solving Quadratic EquationsCompleting the Square

( x + 3)2

x2 + 2(3x) + 9

x2 + 6x

2

6 23

x2 + 6x + 9

3 9

x2 + 6x + 9

( x + 3) ( x + 3)

( x + 3)2

x2 – 14x

2

14 277 49

x2 – 14x + 49

( x – 7) ( x – 7)

( x – 7)2


x2 + 9x

2

9

2

2

94

81

x2 – 5x

4

8192 xx

2

9

2

9xx

2

2

9

x

2

5

2

2

54

25

4

2552 xx

2

5

2

5xx

2

2

5

x


x2 + 8x = 1

2

824 16

1611682 xx

174 2 x

174 2 x

174 x

174 x

4

x2 + 8x = 1


5x2 – 10x + 2 = 0

2

2 21 1

5

5

5

31 x

5

5

5

21 2 x

5

31 2 x

5

31 x

5

31x

1

5x2 – 10x = –2

5

2

5

10

5

5 2

xx

5

222 xx

15

2122 xx

5

31 2 x

5

151x

5

155x

or


2x2 – 2x + 7 = 0

2

1

2

2

1

4

1

2

13

2

1 ix 4

1

4

14

2

12

x

4

13

2

12

x

4

13

2

1 x

2

13

2

1 x

2

1

2x2 – 2x = –7

2

7

2

2

2

2 2

xx

2

72 xx

4

1

2

7

4

12 xx

4

13

2

12

x

2

131 ix

or

The quadratic formula is used to solve any quadratic equation.

2 4

2x

cb b a

a

The quadratic formula is:

Standard form of a quadratic equation is:2 0x xba c

8.2 – Solving Quadratic EquationsThe Quadratic Formula

2 4

2x

cb b a

a


02 cbxax

cbxax 2

a

cx

a

bx

a

a 2

a

cx

a

bx

2

a

b

a

b

22

1 2

22

42 a

b

a

b

a

c

a

b

a

bx

a

bx

2

2

2

22

44

a

a

a

c

a

b

a

bx

a

bx

4

4

44 2

2

2

22

2 4

2x

cb b a

a


22

2

2

22

4

4

44 a

ac

a

b

a

bx

a

bx

2

2

2

22

4

4

4 a

acb

a

bx

a

bx

2

2

2

22

4

4

4 a

acb

a

bx

a

bx

2

22

4

4

2 a

acb

a

bx

2

2

4

4

2 a

acb

a

bx

a

acb

a

bx

2

4

2

2

a

acb

a

bx

2

4

2

2

a

acbbx

2

42

The quadratic formula is used to solve any quadratic equation.

2 4

2x

cb b a

a

The quadratic formula is:

Standard form of a quadratic equation is: 2 0x xba c

2 4 8 0x x

a 1 c b4 8

23 5 6 0x x

a 3 c b 5

22 0x x

a 2 c b1 0

2 10x a 1 c b0 106

2 10 0x


2 4

2x

cb b a

a

2 0x xba c

2 3 2 0x x

2x 1x

1x 2x 0

1 0x 2 0x


2 4

2x

cb b a

a

2 0x xba c

2 3 2 0x x a 1 c b 3 2

23 3 1 24

12x

3 9 8

2x

3 1

2x

3 1

2x

3 1

2x

3 1

2x

4

2x

2x

2

2x

1x 3 1

2x


2 4

2x

cb b a

a

2 0x xba c

22 5 0x x

a 2 c b 1 5

24

22

1 521x

1 1 40

4x

1 41

4x


2 4

2x

cb b a

a


44 2 xx

044 2 xx

42

44411 2 x

8

6411 x

8

631 x

8

631 ix

8

391

ix

8

731 ix

ix

8

73

8

1

2 4

2x

cb b a

a

8.2 – Solving Quadratic EquationsThe Quadratic Formula and the Discriminate

The discriminate is the radicand portion of the quadratic formula (b2 – 4ac).

It is used to discriminate among the possible number and type of solutions a quadratic equation will have.

b2 – 4ac Name and Type of SolutionPositive

Zero

Negative

Two real solutions

One real solutions

Two complex, non-real solutions

2 4

2x

cb b a

a


2143 2

89


Zero

Negative

Two real solutions

One real solutions


2 3 2 0x x a 1 c b 3 2

1

Positive

Two real solutions

2x 1x

2 4

2x

cb b a

a


4441 2

641


Zero

Negative

Two real solutions

One real solutions


a c b

63

Negative


044 2 xx

4 1 4

ix8

73

8

1

2 4

2x

cb b a

a


The Quadratic Formula

Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk.

20 feet

x + 2

x

2 4

2x

cb b a

a




20 feet

x + 2

x

The Pythagorean Theorem

a2 + b2 = c2

(x + 2)2 + x2 = 202

x2 + 4x + 4 + x2 = 400

2x2 + 4x + 4 = 400

2x2 + 4x – 369 = 0

2(x2 + 2x – 198) = 0

2 4

2x

cb b a

a




20 feet

x + 2

x


a2 + b2 = c2

2(x2 + 2x – 198) = 0

12

1981422 2 x

2

79242 x

2

7962 x

2 4

2x

cb b a

a




20 feet

x + 2

x


a2 + b2 = c2

2

7962x

2

2.282

2

2.282 x

2

2.282 x

2

2.26x

1.13x

2

2.30x

1.15xft

2 4

2x

cb b a

a




20 feet

x + 2

x


a2 + b2 = c2

1.13x

ft2.28

ft

21.131.13

28 – 20 = 8 ft

TEMPLATE 1 PPT.ppt

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