#### Transcript 5.2 Solving Quadratic Equations Learning Objective: To solve quadratic equations by taking square roots and to use the Pythagorean Theorem to solve.

Slide 1

Learning Objective: To solve quadratic equations by taking square roots and
to use the Pythagorean Theorem to solve problems involving right triangles.

Warm-up (IN)

CSAP Constructed Response

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

Notes
Properties of Square roots

** Product property -  ab   a  b
 32   16  2  4 2

** Quotient property -  a   a

b
b
4
2
4
 
 

3
9
9

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

EX 1 – Solve

a. 7 x 2  23  103

7 x  126
2

x  18
x   9 2   3 2 or  4.24
2

b. 64x  5  289
2
289

x  5
64
17
x5  
8
2

23
17
5  
x
8
8
or
57
17
x   5   8
8

Try t his p. 282 (middle and bottom)

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

EX 2 – A helicopter 85 ft above a designated area drops a
boxes of food to a disaster area. The height of the box
2
above the ground can be modeled by h t  16t  85 ,
where t is the time in seconds after it has been released.
After how many seconds will it hit the ground?



0  16t 2  85

 85  16t

2

17
2

t
4

t  2.3sec

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

Pythagorean Theorem!
In a right  , the sum of the squares of the lengths of the
legs is equal to the square of the hypotenuse.

a b  c
2

2

2

c

a
b

EX 3 – Find the unknown length
8.4

a. A

B
5.3

b
C

2

b
8.4  5.3
2

b

28
.
09
70.56
2
98.65  b
2
9.93  b
2

2

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

b. J
k

1.9

K

2.8

L

1.9  k  2.8
3.61  k 2  7.84
k 2  4.23
k  2.06
2

2

2

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

EX 4 – What must be the length of QS so that it contacts
the ground halfway between the base of the tower and the
point where PS contacts the ground?

75  PR 2  1062
5626  PR 2  11236
PR 2  5611
1
QR  PR
PR  74.91
2
2

S

106 ft

75 ft

2

SQ
37.45  75
5625  SQ2
83.83 ft  SQ
2

P

Q

R

2

QR  37.45

Out – Describe the procedure you would use to
2
solve 5x  3  12
Summary – So, I was
thinking…

POW!!
HW – p. 287
#21-43 odd,
51-54

Slide 2

Learning Objective: To solve quadratic equations by taking square roots and
to use the Pythagorean Theorem to solve problems involving right triangles.

Warm-up (IN)

CSAP Constructed Response

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

Notes
Properties of Square roots

** Product property -  ab   a  b
 32   16  2  4 2

** Quotient property -  a   a

b
b
4
2
4
 
 

3
9
9

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

EX 1 – Solve

a. 7 x 2  23  103

7 x  126
2

x  18
x   9 2   3 2 or  4.24
2

b. 64x  5  289
2
289

x  5
64
17
x5  
8
2

23
17
5  
x
8
8
or
57
17
x   5   8
8

Try t his p. 282 (middle and bottom)

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

EX 2 – A helicopter 85 ft above a designated area drops a
boxes of food to a disaster area. The height of the box
2
above the ground can be modeled by h t  16t  85 ,
where t is the time in seconds after it has been released.
After how many seconds will it hit the ground?



0  16t 2  85

 85  16t

2

17
2

t
4

t  2.3sec

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

Pythagorean Theorem!
In a right  , the sum of the squares of the lengths of the
legs is equal to the square of the hypotenuse.

a b  c
2

2

2

c

a
b

EX 3 – Find the unknown length
8.4

a. A

B
5.3

b
C

2

b
8.4  5.3
2

b

28
.
09
70.56
2
98.65  b
2
9.93  b
2

2

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

b. J
k

1.9

K

2.8

L

1.9  k  2.8
3.61  k 2  7.84
k 2  4.23
k  2.06
2

2

2

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

EX 4 – What must be the length of QS so that it contacts
the ground halfway between the base of the tower and the
point where PS contacts the ground?

75  PR 2  1062
5626  PR 2  11236
PR 2  5611
1
QR  PR
PR  74.91
2
2

S

106 ft

75 ft

2

SQ
37.45  75
5625  SQ2
83.83 ft  SQ
2

P

Q

R

2

QR  37.45

Out – Describe the procedure you would use to
2
solve 5x  3  12
Summary – So, I was
thinking…

POW!!
HW – p. 287
#21-43 odd,
51-54

Slide 3

Learning Objective: To solve quadratic equations by taking square roots and
to use the Pythagorean Theorem to solve problems involving right triangles.

Warm-up (IN)

CSAP Constructed Response

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

Notes
Properties of Square roots

** Product property -  ab   a  b
 32   16  2  4 2

** Quotient property -  a   a

b
b
4
2
4
 
 

3
9
9

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

EX 1 – Solve

a. 7 x 2  23  103

7 x  126
2

x  18
x   9 2   3 2 or  4.24
2

b. 64x  5  289
2
289

x  5
64
17
x5  
8
2

23
17
5  
x
8
8
or
57
17
x   5   8
8

Try t his p. 282 (middle and bottom)

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

EX 2 – A helicopter 85 ft above a designated area drops a
boxes of food to a disaster area. The height of the box
2
above the ground can be modeled by h t  16t  85 ,
where t is the time in seconds after it has been released.
After how many seconds will it hit the ground?



0  16t 2  85

 85  16t

2

17
2

t
4

t  2.3sec

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

Pythagorean Theorem!
In a right  , the sum of the squares of the lengths of the
legs is equal to the square of the hypotenuse.

a b  c
2

2

2

c

a
b

EX 3 – Find the unknown length
8.4

a. A

B
5.3

b
C

2

b
8.4  5.3
2

b

28
.
09
70.56
2
98.65  b
2
9.93  b
2

2

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

b. J
k

1.9

K

2.8

L

1.9  k  2.8
3.61  k 2  7.84
k 2  4.23
k  2.06
2

2

2

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

EX 4 – What must be the length of QS so that it contacts
the ground halfway between the base of the tower and the
point where PS contacts the ground?

75  PR 2  1062
5626  PR 2  11236
PR 2  5611
1
QR  PR
PR  74.91
2
2

S

106 ft

75 ft

2

SQ
37.45  75
5625  SQ2
83.83 ft  SQ
2

P

Q

R

2

QR  37.45

Out – Describe the procedure you would use to
2
solve 5x  3  12
Summary – So, I was
thinking…

POW!!
HW – p. 287
#21-43 odd,
51-54

Slide 4

Learning Objective: To solve quadratic equations by taking square roots and
to use the Pythagorean Theorem to solve problems involving right triangles.

Warm-up (IN)

CSAP Constructed Response

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

Notes
Properties of Square roots

** Product property -  ab   a  b
 32   16  2  4 2

** Quotient property -  a   a

b
b
4
2
4
 
 

3
9
9

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

EX 1 – Solve

a. 7 x 2  23  103

7 x  126
2

x  18
x   9 2   3 2 or  4.24
2

b. 64x  5  289
2
289

x  5
64
17
x5  
8
2

23
17
5  
x
8
8
or
57
17
x   5   8
8

Try t his p. 282 (middle and bottom)

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

EX 2 – A helicopter 85 ft above a designated area drops a
boxes of food to a disaster area. The height of the box
2
above the ground can be modeled by h t  16t  85 ,
where t is the time in seconds after it has been released.
After how many seconds will it hit the ground?



0  16t 2  85

 85  16t

2

17
2

t
4

t  2.3sec

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

Pythagorean Theorem!
In a right  , the sum of the squares of the lengths of the
legs is equal to the square of the hypotenuse.

a b  c
2

2

2

c

a
b

EX 3 – Find the unknown length
8.4

a. A

B
5.3

b
C

2

b
8.4  5.3
2

b

28
.
09
70.56
2
98.65  b
2
9.93  b
2

2

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

b. J
k

1.9

K

2.8

L

1.9  k  2.8
3.61  k 2  7.84
k 2  4.23
k  2.06
2

2

2

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

EX 4 – What must be the length of QS so that it contacts
the ground halfway between the base of the tower and the
point where PS contacts the ground?

75  PR 2  1062
5626  PR 2  11236
PR 2  5611
1
QR  PR
PR  74.91
2
2

S

106 ft

75 ft

2

SQ
37.45  75
5625  SQ2
83.83 ft  SQ
2

P

Q

R

2

QR  37.45

Out – Describe the procedure you would use to
2
solve 5x  3  12
Summary – So, I was
thinking…

POW!!
HW – p. 287
#21-43 odd,
51-54

Slide 5

Learning Objective: To solve quadratic equations by taking square roots and
to use the Pythagorean Theorem to solve problems involving right triangles.

Warm-up (IN)

CSAP Constructed Response

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

Notes
Properties of Square roots

** Product property -  ab   a  b
 32   16  2  4 2

** Quotient property -  a   a

b
b
4
2
4
 
 

3
9
9

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

EX 1 – Solve

a. 7 x 2  23  103

7 x  126
2

x  18
x   9 2   3 2 or  4.24
2

b. 64x  5  289
2
289

x  5
64
17
x5  
8
2

23
17
5  
x
8
8
or
57
17
x   5   8
8

Try t his p. 282 (middle and bottom)

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

EX 2 – A helicopter 85 ft above a designated area drops a
boxes of food to a disaster area. The height of the box
2
above the ground can be modeled by h t  16t  85 ,
where t is the time in seconds after it has been released.
After how many seconds will it hit the ground?



0  16t 2  85

 85  16t

2

17
2

t
4

t  2.3sec

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

Pythagorean Theorem!
In a right  , the sum of the squares of the lengths of the
legs is equal to the square of the hypotenuse.

a b  c
2

2

2

c

a
b

EX 3 – Find the unknown length
8.4

a. A

B
5.3

b
C

2

b
8.4  5.3
2

b

28
.
09
70.56
2
98.65  b
2
9.93  b
2

2

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

b. J
k

1.9

K

2.8

L

1.9  k  2.8
3.61  k 2  7.84
k 2  4.23
k  2.06
2

2

2

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

EX 4 – What must be the length of QS so that it contacts
the ground halfway between the base of the tower and the
point where PS contacts the ground?

75  PR 2  1062
5626  PR 2  11236
PR 2  5611
1
QR  PR
PR  74.91
2
2

S

106 ft

75 ft

2

SQ
37.45  75
5625  SQ2
83.83 ft  SQ
2

P

Q

R

2

QR  37.45

Out – Describe the procedure you would use to
2
solve 5x  3  12
Summary – So, I was
thinking…

POW!!
HW – p. 287
#21-43 odd,
51-54

Slide 6

Learning Objective: To solve quadratic equations by taking square roots and
to use the Pythagorean Theorem to solve problems involving right triangles.

Warm-up (IN)

CSAP Constructed Response

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

Notes
Properties of Square roots

** Product property -  ab   a  b
 32   16  2  4 2

** Quotient property -  a   a

b
b
4
2
4
 
 

3
9
9

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

EX 1 – Solve

a. 7 x 2  23  103

7 x  126
2

x  18
x   9 2   3 2 or  4.24
2

b. 64x  5  289
2
289

x  5
64
17
x5  
8
2

23
17
5  
x
8
8
or
57
17
x   5   8
8

Try t his p. 282 (middle and bottom)

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

EX 2 – A helicopter 85 ft above a designated area drops a
boxes of food to a disaster area. The height of the box
2
above the ground can be modeled by h t  16t  85 ,
where t is the time in seconds after it has been released.
After how many seconds will it hit the ground?



0  16t 2  85

 85  16t

2

17
2

t
4

t  2.3sec

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

Pythagorean Theorem!
In a right  , the sum of the squares of the lengths of the
legs is equal to the square of the hypotenuse.

a b  c
2

2

2

c

a
b

EX 3 – Find the unknown length
8.4

a. A

B
5.3

b
C

2

b
8.4  5.3
2

b

28
.
09
70.56
2
98.65  b
2
9.93  b
2

2

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

b. J
k

1.9

K

2.8

L

1.9  k  2.8
3.61  k 2  7.84
k 2  4.23
k  2.06
2

2

2

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

EX 4 – What must be the length of QS so that it contacts
the ground halfway between the base of the tower and the
point where PS contacts the ground?

75  PR 2  1062
5626  PR 2  11236
PR 2  5611
1
QR  PR
PR  74.91
2
2

S

106 ft

75 ft

2

SQ
37.45  75
5625  SQ2
83.83 ft  SQ
2

P

Q

R

2

QR  37.45

Out – Describe the procedure you would use to
2
solve 5x  3  12
Summary – So, I was
thinking…

POW!!
HW – p. 287
#21-43 odd,
51-54

Slide 7

Learning Objective: To solve quadratic equations by taking square roots and
to use the Pythagorean Theorem to solve problems involving right triangles.

Warm-up (IN)

CSAP Constructed Response

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

Notes
Properties of Square roots

** Product property -  ab   a  b
 32   16  2  4 2

** Quotient property -  a   a

b
b
4
2
4
 
 

3
9
9

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

EX 1 – Solve

a. 7 x 2  23  103

7 x  126
2

x  18
x   9 2   3 2 or  4.24
2

b. 64x  5  289
2
289

x  5
64
17
x5  
8
2

23
17
5  
x
8
8
or
57
17
x   5   8
8

Try t his p. 282 (middle and bottom)

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

EX 2 – A helicopter 85 ft above a designated area drops a
boxes of food to a disaster area. The height of the box
2
above the ground can be modeled by h t  16t  85 ,
where t is the time in seconds after it has been released.
After how many seconds will it hit the ground?



0  16t 2  85

 85  16t

2

17
2

t
4

t  2.3sec

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

Pythagorean Theorem!
In a right  , the sum of the squares of the lengths of the
legs is equal to the square of the hypotenuse.

a b  c
2

2

2

c

a
b

EX 3 – Find the unknown length
8.4

a. A

B
5.3

b
C

2

b
8.4  5.3
2

b

28
.
09
70.56
2
98.65  b
2
9.93  b
2

2

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

b. J
k

1.9

K

2.8

L

1.9  k  2.8
3.61  k 2  7.84
k 2  4.23
k  2.06
2

2

2

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

EX 4 – What must be the length of QS so that it contacts
the ground halfway between the base of the tower and the
point where PS contacts the ground?

75  PR 2  1062
5626  PR 2  11236
PR 2  5611
1
QR  PR
PR  74.91
2
2

S

106 ft

75 ft

2

SQ
37.45  75
5625  SQ2
83.83 ft  SQ
2

P

Q

R

2

QR  37.45

Out – Describe the procedure you would use to
2
solve 5x  3  12
Summary – So, I was
thinking…

POW!!
HW – p. 287
#21-43 odd,
51-54

Slide 8

Learning Objective: To solve quadratic equations by taking square roots and
to use the Pythagorean Theorem to solve problems involving right triangles.

Warm-up (IN)

CSAP Constructed Response

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

Notes
Properties of Square roots

** Product property -  ab   a  b
 32   16  2  4 2

** Quotient property -  a   a

b
b
4
2
4
 
 

3
9
9

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

EX 1 – Solve

a. 7 x 2  23  103

7 x  126
2

x  18
x   9 2   3 2 or  4.24
2

b. 64x  5  289
2
289

x  5
64
17
x5  
8
2

23
17
5  
x
8
8
or
57
17
x   5   8
8

Try t his p. 282 (middle and bottom)

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

EX 2 – A helicopter 85 ft above a designated area drops a
boxes of food to a disaster area. The height of the box
2
above the ground can be modeled by h t  16t  85 ,
where t is the time in seconds after it has been released.
After how many seconds will it hit the ground?



0  16t 2  85

 85  16t

2

17
2

t
4

t  2.3sec

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

Pythagorean Theorem!
In a right  , the sum of the squares of the lengths of the
legs is equal to the square of the hypotenuse.

a b  c
2

2

2

c

a
b

EX 3 – Find the unknown length
8.4

a. A

B
5.3

b
C

2

b
8.4  5.3
2

b

28
.
09
70.56
2
98.65  b
2
9.93  b
2

2

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

b. J
k

1.9

K

2.8

L

1.9  k  2.8
3.61  k 2  7.84
k 2  4.23
k  2.06
2

2

2

Learning Objective: To solve quadratic equations by taking square roots and to use the
Pythagorean Theorem to solve problems involving right triangles.

EX 4 – What must be the length of QS so that it contacts
the ground halfway between the base of the tower and the
point where PS contacts the ground?

75  PR 2  1062
5626  PR 2  11236
PR 2  5611
1
QR  PR
PR  74.91
2
2

S

106 ft

75 ft

2

SQ
37.45  75
5625  SQ2
83.83 ft  SQ
2

P

Q

R

2

QR  37.45

Out – Describe the procedure you would use to
2
solve 5x  3  12
Summary – So, I was
thinking…

POW!!
HW – p. 287
#21-43 odd,
51-54