1. √49 2. –√144 Lesson 4.5, For use with pages 266-271 Find the exact value. 1. √49 ANSWER 7 2. –√144 ANSWER –12

3. Use calculator to approximate the value of to the nearest tenth. Lesson 4.5, For use with pages 266-271 16 3. Use calculator to approximate the value of to the nearest tenth. 82 ANSWER 2.3 4. The area of half of a square mural is 60 square feet. What is the length of a side of the mural? 1 2 ANSWER 11 ft

4.5 Properties of Square Roots

EXAMPLE 1 Use properties of square roots Simplify the expression. a. 80 5 16 = 5 = 4 b. 6 21 126 = 9 14 = = 3 14 c. 4 81 = 4 81 = 2 9 d. 7 16 = 7 16 = 4 7

GUIDED PRACTICE GUIDED PRACTICE for Example 1 27 SOLUTION 27 3 9 = 3 = 3 98 SOLUTION 98 2 49 = 2 = 7

GUIDED PRACTICE GUIDED PRACTICE for Example 1 10 15 SOLUTION 10 15 6 25 150 = 6 = 5 8 28 SOLUTION 8 28 224 = 14 16 = 14 = 4

GUIDED PRACTICE GUIDED PRACTICE for Example 1 9 64 SOLUTION 9 64 9 64 = = 3 8 15 4 SOLUTION 15 4 15 4 = = 15 2

GUIDED PRACTICE GUIDED PRACTICE for Example 1 11 25 SOLUTION 11 25 11 25 = = 5 11 36 49 SOLUTION 36 49 36 49 = 7 6 =

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 =

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

Write original 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.

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 

Write original equation. 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

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

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

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

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

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

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

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

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

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

Write original equation. 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.

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 

Write original equation. 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.

GUIDED PRACTICE for Examples 2, 3, and 4 ANSWER The solutions are and . 6 – 6 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 

Write original equation. GUIDED PRACTICE for Examples 2, 3, and 4 3(x – 2)2 = 40 SOLUTION 3(x – 2)2 = 40 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 +

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

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 ?

Substitute 0 for h and 50 for h0. 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.

Substitute 0 for h and 30 for h0. GUIDED PRACTICE GUIDED PRACTICE for Example 5 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? 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 GUIDED PRACTICE for Example 5 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.