10.4

Warm UpWarm Up

Lesson QuizLesson Quiz

Lesson PresentationLesson Presentation

Use Square Roots to Solve Quadratic Equations

10.4 Warm-Up

Evaluate the expression.

ANSWER –7

ANSWER about 14.1

1. – 49√

2. 200√

3. ± 121√

ANSWER ±11

10.4 Warm-Up

ANSWER 0.75 sec

A ball is dropped from a height 9 feet above the ground. How long does it take the ball to hit the ground?

4.

10.4 Example 1

Solve the equation.

a. 2x2 = 8

SOLUTION

a. 2x2 = 8 Write original equation.

x2 = 4 Divide each side by 2.

x = ± 4 = ± 2 Take square roots of each side. Simplify.

The solutions are –2 and 2.ANSWER

b. m2 – 18 = – 18 c. b2 + 12 = 5

10.4 Example 1

b. m2 – 18 = – 18 Write original equation.

m2 = 0 Add 18 to each side.

The square root of 0 is 0.m = 0

ANSWER

The solution is 0.

10.4 Example 1

c. b2 + 12 = 5 Write original equation.

b2 = –7 Subtract 12 from each side.

ANSWER

Negative real numbers do not have real square roots. So, there is no solution.

10.4 Example 2

Solve 4z2 = 9.

SOLUTION

4z2 = 9 Write original equation.

z2 = 94 Divide each side by 4.

Take square roots of each side.z = ± 94

z = ± 32

Simplify.

The solutions are – and . 32

32

ANSWER

10.4 Guided Practice

Solve the equation.

1. c2 – 25 = 0 ANSWER –5, 5.

2. 5w2 + 12 = – 8 ANSWER no solution

3. 2x2 + 11 = 11 ANSWER 0

4. 25x2 = 16 ANSWER 4 5

4 5– ,

5. 9m2 = 100 ANSWER 103– ,

103

6. 49b2 + 64 = 0 ANSWER no solution

10.4 Example 3

Solve 3x2 – 11 = 7. Round the solutions to the nearesthundredth.

SOLUTION

3x2 – 11 = 7 Write original equation.

3x2 = 18 Add 11 to each side.

x2 = 6 Divide each side by 3.

x = ± 6 Take square roots of each side.

x ± 2.45 Use a calculator. Round to the nearesthundredth.

ANSWER

The solutions are about – 2.45 and about 2.45.

10.4 Guided Practice

Solve the equation. Round the solutions to the nearest hundredth.

7. x2 + 4 = 14 ANSWER – 3.16, 3.16

8. 3k2 – 1 = 0 ANSWER – 0.58, 0.58

9. 2p2 – 7 = 2 ANSWER – 2.12, 2.12

10.4 Example 4

Solve 6(x – 4)2 = 42. Round the solutions to the nearesthundredth.

6(x – 4)2 = 42 Write original equation.

(x – 4)2 = 7 Divide each side by 6.

x – 4 = ± 7 Take square roots of each side.

7 x = 4 ± Add 4 to each side.

ANSWER

The solutions are 4 + 6.65 and 4 – 1.35.7 7

10.4 Example 4

CHECKTo check the solutions, first write the equation so that 0 is on one side as follows: 6(x – 4)2 – 42 = 0. Then graph the related function y = 6(x – 4)2 – 42. The x-intercepts appear to be about 6.6 and about 1.3. So, each solution checks.

10.4 Guided Practice

Solve the equation. Round the solution to the nearest hundredth if necessary.

10. 2(x – 2)2 = 18 ANSWER –1, 5

11. 4(q – 3)2 = 28 ANSWER 0.35, 5.65

12. 3(t + 5)2 = 24 ANSWER –7.83, –2.17

10.4 Example 5

During an ice hockey game, a remote-controlled blimp flies above the crowd and drops a numbered table-tennis ball. The number on the ball corresponds to a prize. Use the information in the diagram to find the amount of time that the ball is in the air.

SPORTS EVENT

10.4 Example 5

STEP 1Use the vertical motion model to write an equation for the height h (in feet) of the ball as a function of time t (in seconds).

SOLUTION

h = –16t2 + vt + s Vertical motion model

h = –16t2 + 0t + 45 Substitute for v and s.

STEP 2Find the amount of time the ball is in the air by substituting 17 for h and solving for t.

10.4 Example 5

Write model.

Substitute 17 for h.

Subtract 45 from each side.

Divide each side by 16.

Take positive square root.

Use a calculator.

h = –16t2 + 45

17 = –16t2 + 45

– 28 = –16t2

28 16

= t2

28 16

= t

1.32 t

ANSWER

The ball is in the air for about 1.32 seconds

10.4 Guided Practice

WHAT IF? In Example 5, suppose the table-tennis ball is released 58 feet above the ground and is caught 12 feet above the ground. Find the amount of time that the ball is in the air. Round your answer to the nearesthundredth of a second.

13.

ANSWER

The ball is in the are for about 1.70 second.

10.4 Lesson Quiz

ANSWER –2, 2

Solve the equation. Round solutions to the nearest hundredth, if necessary.

1. 4b2 – 13 = 3

2. 9x2 = 25

ANSWER –3.16, 3.16

ANSWER – 53

53

,

3. 3n2 –18 = 12

10.4 Lesson Quiz

At a football game you are sitting 32 feet above the ground. If your hat comes off and falls to the ground, how long will it be in the air?

4.

ANSWER About 1.41 sec