lesson 4.5, for use with pages 266-271 find the exact value. answer 7 –12 1.√49 2.–√144
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Lesson 4.5, For use with pages 266-271
Find the exact value.
ANSWER 7
ANSWER –12
1. √49
2. –√144
Lesson 4.5, For use with pages 266-271
ANSWER 2.3
ANSWER 11 ft
4. The area of half of a square mural is 60 square feet. What is the length of a side of the mural?
12
163. Use calculator to approximate the value of
to the nearest tenth.
82
4.5 Properties of Square Roots
EXAMPLE 1 Use properties of square roots
Simplify the expression.
5= 4a.
80 516=
= 3 14b.
6 21 126= 9 14=
c.
4
81=
4
81=
2 9
d.
7
16=
7
16=
4
7
GUIDED PRACTICEGUIDED PRACTICE for Example 1
271.
3= 327 39=
SOLUTION
982.
2= 798 249=
SOLUTION
GUIDED PRACTICEGUIDED PRACTICE for Example 1
3.
6= 5
SOLUTION
SOLUTION
10 15
10 15 625150 ==
4. 8 28
14= 41416=224= 8 28
GUIDED PRACTICEGUIDED PRACTICE for Example 1
5.
SOLUTION
SOLUTION
9
64=
3 8
9
64
9
64=
6. 15
4
15
4
15
4= =
15
2
GUIDED PRACTICEGUIDED PRACTICE for Example 1
7.
SOLUTION
SOLUTION
11
25
11
25
11
25=
8. 36
49
36
4936
49=
= 5
11
76
=
EXAMPLE 2 Rationalize denominators of fractions.
Simplify (a) 5
2and 3
7 + 2
SOLUTION
(a) 5
2
2
10=
=5
2
=5
2
2
2
(b)
EXAMPLE 2 Rationalize denominators of fractions.
SOLUTION
(b)3
7 + 2=
3
7 + 2 7 – 2
7 – 2
=21 – 3 2
49 – 7 + 7 – 2 2 2
=21 – 3 2
47
EXAMPLE 3 Solve a quadratic equation
Solve 3x2 + 5 = 41.
3x2 + 5 = 41 Write original equation.
3x2 = 36 Subtract 5 from each side.
x2 = 12 Divide each side by 3.
x = + 12 Take square roots of each side.
x = + 4 3
x = + 2 3
Product property
Simplify.
EXAMPLE 3 Solve a quadratic equation
ANSWER
The solutions are and 2 3 2 3–
Check the solutions by substituting them into the original equation.
3x2 + 5 = 41
3( )2 + 5 = 41 ?2 3
41 = 41
3(12) + 5 = 41 ?
3x2 + 5 = 41
3( )2 + 5 = 41 ? – 2 3
41 = 41
3(12) + 5 = 41 ?
EXAMPLE 4 Standardized Test Practice
SOLUTION
15 (z + 3)2 = 7 Write original equation.
(z + 3)2 = 35 Multiply each side by 5.
z + 3 = + 35 Take square roots of each side.
z = – 3 + 35 Subtract 3 from each side.
The solutions are – 3 + and – 3 – 35 35
EXAMPLE 4 Standardized Test Practice
ANSWER
The correct answer is C.
GUIDED PRACTICE for Examples 2, 3, and 4GUIDED PRACTICE
Simplify the expression.
9.
SOLUTION
6
5
5
30=
=6
5
=6
5
5
5
6
5
GUIDED PRACTICE for Examples 2, 3, and 4GUIDED PRACTICE
Simplify the expression.
10.
SOLUTION
9
8=
9
8
=9
8
8
8
9
8
2
4=
3
GUIDED PRACTICE for Examples 2, 3, and 4GUIDED PRACTICE
Simplify the expression.
11.
SOLUTION
17
12=
17
12
=17
12
12
12
17
12
51
12=
2 51
6=
GUIDED PRACTICE for Examples 2, 3, and 4GUIDED PRACTICE
Simplify the expression.
12.
SOLUTION
19
21
399
21=
=19
21
=19
21
21
21
19
21
SOLUTION
– 6
7 – 5=
– 6
7 – 5 7 + 5
7 + 5
=– 42 – 6 5
49 – 7 + 7 – 5 5 5
=– 21 – 3 5
22
for Examples 2, 3, and 4GUIDED PRACTICE
13. – 6
7 – 5
SOLUTION
2
4 + 11=
2
4 + 11 4 – 11
4 – 11
=8 – 2 11
16 – 4 + 4 – 1111 11
=8 – 2 11
5
for Examples 2, 3, and 4GUIDED PRACTICE
14. 2
4 + 11
SOLUTION
– 1
9 + 7=
– 1
9 + 7 9 – 7
9 – 7
=– 9 + 7
81 – 9 + 9 – 77 7
=– 9 + 7
74
for Examples 2, 3, and 4GUIDED PRACTICE
15. – 1
9 + 7
SOLUTION
4
8 – 3=
4
8 – 3 8 + 3
8 + 3
=32 + 4 3
64 – 4 + 4 – 3 3 3
=32 + 4 3
61
for Examples 2, 3, and 4GUIDED PRACTICE
16. 4
8 – 3
SOLUTION
for Examples 2, 3, and 4GUIDED PRACTICE
17.
Solve the equation.
5x2 = 80
Write original equation.5x2 = 80
x2 = 16 Divide each side by 5.
x = + 16 Take square roots of each side.
x = + 4 4
x = + 4
Product property
Simplify.
for Examples 2, 3, and 4GUIDED PRACTICE
ANSWER
The solutions are and . 4 4–
Check Check the solutions by substituting them into the original equation.
5x2 = 80
5(– 4)2 = 80 ?
80 = 80
5(16) = 80?
5x2 = 80
80 = 80
5(16) = 80?
5(4)2 = 80 ?
SOLUTION
for Examples 2, 3, and 4GUIDED PRACTICE
18.
Solve the equation.
z2 – 7 = 29
Write original equation.
z2 = 36
z = + 36 Take square roots of each side.
z = + 6 6
z = + 6
Product property
Simplify.
z2 – 7 = 29
Add 7 to each side.
for Examples 2, 3, and 4GUIDED PRACTICE
ANSWER
The solutions are and . 6 6–
Check the solutions by substituting them into the original equation.
29 = 29
z2 – 7 = 29
(6)2 – 7 = 29 ?
?36 – 7 = 2929 = 29
z2 – 7 = 29
(– 6)2 – 7 = 29 ?
36 – 7 = 29 ?
SOLUTION
for Examples 2, 3, and 4GUIDED PRACTICE
19.
Write original equation.
Take square roots of each side.
Division property
3(x – 2)2 = 40
3(x – 2)2 = 40
Divide each side by 3.(x – 2)2 =403
+(x – 2) =403
x =40
32 +
x =40
32 + 3
3
=120
32 +
for Examples 2, 3, and 4GUIDED PRACTICE
=4(30)
32 +
= 2 30
32 +
ANSWER
The solutions are . 2 30
32 +
EXAMPLE 5 Model a dropped object with a quadratic function
Science Competition
For a science competition, students must design a container that prevents an egg from breaking when dropped from a height of 50 feet. How long does the container take to hit the ground ?
EXAMPLE 5 Model a dropped object with a quadratic function
SOLUTION
h = – 16t 2 + h0 Write height function. 0 = – 16t 2 + 50 Substitute 0 for h and 50 for h0.
– 50 = – 16t 2 Subtract 50 from each side.5016 = t2 Divide each side by – 16.
50
16+ = t2 Take square roots of each side.
+ 1.8 t Use a calculator.
EXAMPLE 5 Model a dropped object with a quadratic function
ANSWER
Reject the negative solution, – 1.8 , because time must be positive. The container will fall for about 1.8 seconds before it hits the ground.
GUIDED PRACTICE for Example 5GUIDED PRACTICE
What If? In Example 5, suppose the egg container is dropped from a height of 30 feet. How long does the container take to hit the ground?
20.
SOLUTION
h = – 16t 2 + h0 Write height function. 0 = – 16t 2 + 30 Substitute 0 for h and 30 for h0.
– 30 = – 16t 2 Subtract 30 from each side.3016 = t2 Divide each side by – 16.
30
16+ = t2 Take square roots of each side.
+ 1.4 t Use a calculator.
GUIDED PRACTICE for Example 5GUIDED PRACTICE
ANSWER
Reject the negative solution, – 1.4, because time must be positive. The container will fall for about 1.4 seconds before it hits the ground.
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