1. EXAMPLE 2
Rationalize denominators of fractions. 5 2 3
7 + 2
Simplify
(a)
and
(b)
SOLUTION
(a) 5 2 = 5 2 = 5 2 2 10 =

2. EXAMPLE 2
Rationalize denominators of fractions.
SOLUTION = 3
7 + 2
7 –
(b) 3
7 + 2 =
21 – 3 2
49 – – 2 =
21 – 3 2 47

3. Solve a quadratic equation
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
Product property
x =
+ 2 3
Simplify.

4. Solve a quadratic equation
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
3x2 + 5 = 41
3( )2 + 5 = 41 2 3 ?
3( )2 + 5 = 41
– 2 3 ?
3(12) + 5 = 41 ?
3(12) + 5 = 41 ?
41 = 41 
41 = 41 

5. Standardized Test Practice
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 – and – 3 – 35

6. EXAMPLE 4
Standardized Test Practice
ANSWER
The correct answer is C.

7. GUIDED PRACTICE
GUIDED PRACTICE
for Examples 2, 3, and 4
Simplify the expression. 6 5
SOLUTION 6 5 = 6 5 = 6 5 5 30 =

8. GUIDED PRACTICE
GUIDED PRACTICE
for Examples 2, 3, and 4
Simplify the expression. 9 8
SOLUTION 9 8 = 9 8 = 9 8 2 4 = 3

9. GUIDED PRACTICE
GUIDED PRACTICE
for Examples 2, 3, and 4
Simplify the expression. 17 12
SOLUTION 17 12 = 17 12 = 17 12 51 12 = 2 6

10. GUIDED PRACTICE
GUIDED PRACTICE
for Examples 2, 3, and 4
Simplify the expression. 19 21
SOLUTION 19 21 = 19 21 = 19 21
399 21 =

11. GUIDED PRACTICE
for Examples 2, 3, and 4
– 6
7 – 5
SOLUTION =
– 6
7 – 5
7 +
– 6
7 – 5 =
– 42 – 6 5
49 – – 5 =
– 21 – 3 5 22

12. GUIDED PRACTICE
for Examples 2, 3, and 4 2
4 + 11
SOLUTION = 2
4 + 11
4 – 2
4 + 11 =
8 – 2 11
16 – – 11 =
8 – 2 11 5

13. GUIDED PRACTICE
for Examples 2, 3, and 4
– 1
9 + 7
SOLUTION
– 1
9 + 7 =
9 – =
– 9 + 7
81 – – 7 =
– 9 + 7 74

14. GUIDED PRACTICE
for Examples 2, 3, and 4 4
8 – 3
SOLUTION = 4
8 – 3
8 + 4
8 – 3 =
32 + 4 3
64 – – 3 =
32 + 4 3 61

15. GUIDED PRACTICE for Examples 2, 3, and 4 Solve the equation. 5x2 = 80
SOLUTION
5x2 = 80
Write original equation.
x2 = 16
Divide each side by 5.
x = + 16
Take square roots of each side.
x = + 4
Product property
x =
+ 4
Simplify.

16. GUIDED PRACTICE for Examples 2, 3, and 4 ANSWER
The solutions are and . 4 – 4
Check Check the solutions by substituting them into the original equation.
5x2 = 80
5x2 = 80
5(4)2 = 80 ?
5(– 4)2 = 80 ?
5(16) = 80 ?
5(16) = 80 ?
80 = 80 
80 = 80 

17. GUIDED PRACTICE for Examples 2, 3, and 4 Solve the equation.
z2 – 7 = 29
SOLUTION
z2 – 7 = 29
Write original equation.
z2 = 36
Add 7 to each side.
z = + 36
Take square roots of each side.
z = + 6
Product property
z =
+ 6
Simplify.

18. GUIDED PRACTICE for Examples 2, 3, and 4 ANSWER
The solutions are and . 6 – 6
Check Check the solutions by substituting them into the original equation.
z2 – 7 = 29
z2 – 7 = 29 ?
(6)2 – 7 = 29 ?
(– 6)2 – 7 = 29 ?
36 – 7 = 29 ?
36 – 7 = 29
29 = 29 
29 = 29 

19. GUIDED PRACTICE for Examples 2, 3, and 4 3(x – 2)2 = 40 SOLUTION
Write original equation.
(x – 2)2 = 40 3
Divide each side by 3. +
(x – 2) = 40 3
Take square roots of each side.
x = 40 3 2 +
Division property
x = 40 3 2 + =
120 3 2 +

20. GUIDED PRACTICE
for Examples 2, 3, and 4 =
4(30) 3 2 + =
2 30 3 2 +
ANSWER
The solutions are
2 30 3 2 +