Square Roots and Radicals

• Try some:
• Simplify these:
1
25
0




 144

16

5

1
0

12

• More examples:
144

12

48

196

4 3

13


2
x
2 
4 x
2 
4 x

2 or 2 *
* Remember, the square root of a positive # has
2 answers! (one + and one -)

1.
Solve. 3 - 5x 2 = -9
-3 -3
-5x 2 = -12
-5 -5 x 2 = 12
5 x
2 
12
5
2. Solve. 3(x-2) 2 =21
3 3
(x-2) 2 = 7
( x

2 )
2  
7 x

2
  x

2

7
7 x

12

5
12 * 5

5 * 5
60
25

4 * 15
5


2
5
15

3.
Solve. 4x 2 -6=42
+6 +6
4x 2 =48
4 4 x 2 = 12 x
2  
12 x
 
4 * 3
 
2 3
4. Solve.
1
5
( x

4 )
2
( x

4 )
2

6

30
( x

4 )
2   x

4
  x

4

30
30
30

You CANNOT leave a radical in the denominator of a fraction!
No tents in the basement!!!!
(the numerator is OK)
Just multiply the top & bottom of the fraction by the radical to “rationalize” the denominator.

(a>0 and b>0)
1.
Product Property – ab
 a * b
2.
Quotient Propertya b
 a b
Example:
40

4 * 10

4 * 10

2 10
Example:
3
4

3
4

2
3

18


9

2
9

2

3 2


16

6

16

6

4 6
96

12

4

3

4

3

2 3
45

9

5

9

5

3 5


 
2

5

5

5

5


5
25



7

7

7

7


7
49
 
2


 
2

10

10

10

10


10
100

3 7

4 2

3

4 7


12 7

2
2

12 14

6 3

5 3

6

5 3


30 3

3
3

30

30

3
9

90

8 2

5 10

8

5 2


40 2

10

40 20

40

4

10
5

40

2

5

80 5

7 14

5 2

7

5 14


35 14

2

35 28

35

4

7

35

2

7
2

70 7

A.
Simplify each expression.
Find a perfect square factor of 32.
Product Property of Square Roots
B.
Quotient Property of Square Roots

C
Simplify each expression.
Product Property of Square Roots
D.
Quotient Property of Square Roots

E.
Simplify each expression.
Find a perfect square factor of 48.
Product Property of Square Roots
F.
Quotient Property of Square Roots
Simplify.

G.
Simplify each expression.
Product Property of Square Roots
H.
Quotient Property of Square Roots

500 
100 * 5

10 5

3 12 * 6

3 72

3 36 * 2

3 * 6 2

18
25 
5
3.
9

25
9 3
2

1.
25
3

25
3

5
*
3 *
3
3

5
9
3

5
3
3
Can’t have a tent in the basement!
2.
2
11

2
11
*
*
11
11

22
121

22
11

Simplify by rationalizing the denominator.
Multiply by a form of 1.
= 2

Simplify the expression.
Multiply by a form of 1.

Simplify by rationalizing the denominator.
Multiply by a form of 1.

Simplify by rationalizing the denominator.
Multiply by a form of 1.

3.
4.
5.
1.
Estimate to the nearest tenth.
Simplify each expression.
2.
6.7


Solving Quadratic Equations using the Quadratic
Formula
General equation of a quadratic:
Quadratic
Formula: ax 2  bx
 c

0
Notice where the letters come from for the formula x

 b
 b
2 
4 ac
2 a
You must get equation equal to zero before you determine a, b, and c.
We use the quadratic formula when something can not be factored. However, it also works for factorable quadratic equations as well.

Ex. 1
Solve.
2 x x

(5
2
) (5 4 x
2 x

5

1
2 x 3 c b a

1

 
5
6 x

2

Ex. 2
Solve.
2 x x

 
( )
2 c b a

1
 
2
 
8 x

Ex. 3
Solve.
2
4


2
4 4 4 b c a



1 4
4
1
9 x

Ex. 4
Solve.
2
8

(
2 n o i n b c a



1
1 8
1 0

Ex. 5
Suppose a football player kicks a ball and gives it an initial upward velocity of 47ft/s. The starting height of the football is 3ft. If no one catches the football, how long will it be in the air?
NOTE : For your homework tomorrow night, DO NOT answer the question. I want you to draw a sketch of what you think this picture would look like on a graph. Tell me what the x-axis symbolizes, what the y-axis symbolizes, and what the zeros represent in the problem.
What does the x-axis stand for?
Time the ball is in the air
What does the y-axis stand for?
Height of the ball
What do the zeros represent?
The amount of time it takes the ball to hit the ground.

x 2 
8 x

13

0 a

1 b
 
8 c

13 x

 b
 b
2 
4 ac
2 a x


   
2
 
2 
4
   x

8

2
12
12


2
4

3
3 x

8

2
2
3 x

4

3
Before you can find a,b, and c, you must get equation = to 0.
Type what is under the radical exactly as written on your calculator.
Simplify the radical
Divide by the denominator, if you can
Roots:

3 x 2 
5

0 a

3 b

0 c
 
5 x

 b
 b
2 
4 ac
2 a x


   
2
2
 

4
  
60


2
4

15
15 x

0

6
60 x


2
6
15 x
 
15
3
Solve the quadratic equation by using the quadratic formula and leave answers in simplest radical form.
Simplify the radical
Divide by the denominator, if you can. Since I can’t, divide by GCF
Roots

x 2 
12 x

29

0 a

1 b
 
12 c

29 x
 x

 b
 b
2
2 a

4 ac



12
 

2
12
 

2 
4
   x

12

2 x

12

2
28 x
28

12

2
28 x x


8 .
64
8 .
6 x x


3 .
35
3 .
4
Solve the quadratic equation by using the quadratic formula and round answers to the nearest tenth.
Since we are solving to the nearest tenth, we do not need to simplify the radical!
Be careful when typing this into your calculator. I recommend that you type in the numerator then hit enter, then divide by denominator!
Roots:

x

2 
10 x
8


10
10 x x x
2 
10 x

8

0 a b

1
 
10 x

 b
 b
2
2 a

4 ac c

8 x x





10
 
2

10
 

2 
4
   x

10

2
68
10

2
68 x

10

2
68
Solve the quadratic equation by using the quadratic formula and round answers to the nearest tenth.
x

9 .
12 x

9 .
1 x

.
87 x

.
9
Roots:


3
 
2
 
1
3 x
2


6 x
1


1
1
3 x
2 
6 x

1

0 a

3 b
 
6 c
 
1 x

 b
 b
2 
4 ac
2 a x


   
2
 
2 
4
   x

6

6
48
Solve the quadratic equation by using the quadratic formula and round answers to the nearest tenth.
x

6

6
48 x

2 .
15 x

2 .
1
Roots x

6

6
48 x
 
.
15 x
 
.
2

2 x
2
2 x

2
4
 x
1


4 x
4 x

4 x

1

0 a

2 b
 
4 c

1 x

 b
 b
2 
4 ac
2 a x


   
2
 
2 
4
  
8


2
4

2
2 x x


4

4

4
2
4
8
2
Solve the quadratic equation by using the quadratic formula and leave answers in simplest radical form.
x

2

2
2
Roots:
Divide by the denominator, if you can. Since I can’t, divide by GCF