Warm Up Lesson Presentation Lesson Quiz€¦ · Holt Algebra 1 Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 1. Holt McDougal Algebra 1 8-8 Completing the Square Warm

Holt McDougal Algebra 1

8-8 Completing the Square8-8 Completing the Square

Holt Algebra 1

Warm Up

Lesson Presentation

Lesson Quiz



8-8 Completing the Square

Warm Up

Simplify.

19 1. 2.

3. 4.



Warm Up

Solve each quadratic equation by factoring.

5. x2 + 8x + 16 = 0

6. x2 – 22x + 121 = 0

7. x2 – 12x + 36 = 0

x = –4

x = 11

x = 6



Solve quadratic equations by completing the square.

Objective



completing the square

Vocabulary



In the previous lesson, you solved quadratic equations by isolating x2 and then using square roots. This method works if the quadratic equation, when written in standard form, is a perfect square.

When a trinomial is a perfect square, there is a relationship between the coefficient of the x-term and the constant term.

X2 + 6x + 9 x2 – 8x + 16 Divide the coefficient of

the x-term by 2, then

square the result to get

the constant term.



An expression in the form x2 + bx is not a perfect square. However, you can use the relationship shown above to add a term to x2 + bx to form a trinomial that is a perfect square. This is called completing the square.



Example 1: Completing the Square

Complete the square to form a perfect square trinomial.

A. x2 + 2x + B. x2 – 6x +

x2 + 2x

x2 + 2x + 1

x2 + –6x

x2 – 6x + 9

Identify b.

.



Check It Out! Example 1


a. x2 + 12x + b. x2 – 5x +

x2 + 12x

x2 + 12x + 36

x2 + –5xIdentify b.

x2 – 5x +

.





c. 8x + x2 +

x2 + 8x

x2 + 8x + 16

Identify b.

.



To solve a quadratic equation in the form x2 + bx = c, first complete the square of x2 + bx. Then you can solve using square roots.



Solving a Quadratic Equation by Completing the Square



Example 2A: Solving x2 +bx = c by Completing the

Square

Solve by completing the square. Check your answer.

x2 + 16x = –15

Step 1 x2 + 16x = –15

Step 2

Step 3 x2 + 16x + 64 = –15 + 64

Step 4 (x + 8)2 = 49

Step 5 x + 8 = ± 7

Step 6 x + 8 = 7 or x + 8 = –7

x = –1 or x = –15

The equation is in the

form x2 + bx = c.

Complete the square.

Factor and simplify.

Take the square root

of both sides.

Write and solve two

equations.

.



Example 2A Continued


x2 + 16x = –15

The solutions are –1 and –15.

Check x2 + 16x = –15

(–1)2 + 16(–1) –15

1 – 16 –15

–15 –15✓

x2 + 16x = –15

(–15)2 + 16(–15) –15

225 – 240 –15

–15 –15✓



Example 2B: Solving x2 +bx = c


x2 – 4x – 6 = 0

Step 1 x2 + (–4x) = 6

Step 3 x2 – 4x + 4 = 6 + 4

Step 4 (x – 2)2 = 10

Step 5 x – 2 = ± √10

Write in the form

x2 + bx = c.




of both sides.

Write and solve two

equations.

Step 6 x – 2 = √10 or x – 2 = –√10

x = 2 + √10 or x = 2 – √10

.Step 2



Example 2B Continued


The solutions are 2 + √10 and x = 2 – √10.

Check Use a graphing calculator to check your answer.



Check It Out! Example 2a


x2 + 10x = –9

Step 1 x2 + 10x = –9

Step 3 x2 + 10x + 25 = –9 + 25 Complete the square.

The equation is in the

form x2 + bx = c.

Step 2

Step 4 (x + 5)2 = 16

Step 5 x + 5 = ± 4

Step 6 x + 5 = 4 or x + 5 = –4

x = –1 or x = –9



of both sides.

Write and solve two

equations.

.



Check It Out! Example 2a Continued


x2 + 10x = –9

The solutions are –9 and –1.

x2 + 10x = –9

(–9)2 + 10(–9) –9

81 – 90 –9

–9 –9 ✓

x2 + 10x = –9

(–1)2 + 10(–1) –9

1 – 10 –9

–9 –9 ✓

Check



Check It Out! Example 2b


t2 – 8t – 5 = 0

Step 1 t2 + (–8t) = 5

Step 3 t2 – 8t + 16 = 5 + 16 Complete the square.

Write in the form

x2 + bx = c.

Step 2

Step 4 (t – 4)2 = 21

Step 5 t – 4 = ± √21



of both sides.

Write and solve two

equations.

Step 6 t = 4 + √21 or t = 4 – √21

.



Check It Out! Example 2b Continued


t = 4 – √21 or t = 4 + √21.The solutions are

Check Use a graphing calculator to check your answer.



Example 3A: Solving ax2 + bx = c by Completing the

Square

Solve by completing the square.

–3x2 + 12x – 15 = 0

Step 1

x2 – 4x + 5 = 0x2 – 4x = –5

x2 + (–4x) = –5

Step 3 x2 – 4x + 4 = –5 + 4

Divide by – 3 to make a = 1.

Write in the form x2 + bx = c.


.Step 2



Example 3A Continued


–3x2 + 12x – 15 = 0

Step 4 (x – 2)2 = –1

There is no real number whose square is negative, so there are no real solutions.




Example 3B: Solving ax2 + bx = c by Completing the

Square


5x2 + 19x = 4

Step 1 Divide by 5 to make a = 1.


Step 2 .



Complete the square.Step 3



Factor and simplify.Step 4

Step 5Take the square root

of both sides.

Rewrite using like

denominators.





Step 6

The solutions are and –4.

Write and solve

two equations.



Check It Out! Example 3a


3x2 – 5x – 2 = 0







Step 3

Step 4



Step 2 .





Write and solve two

equations.

Step 6

−


of both sides.

Step 5



Check It Out! Example 3b


4t2 – 4t + 9 = 0





Check It Out! Example 3b Continued


4t2 – 4t + 9 = 0

Step 2

Step 3



Step 4

There is no real number whose square is negative, so there are no real solutions.

.



Example 4: Problem-Solving Application

A rectangular room has an area of 195 square feet. Its width is 2 feet shorter than its length. Find the dimensions of the room. Round to the nearest hundredth of a foot, if necessary.

1 Understand the Problem

The answer will be the length and width of the room.

List the important information:

• The room area is 195 square feet.

• The width is 2 feet less than the length.



2 Make a Plan

Example 4 Continued

Set the formula for the area of a rectangle equal to 195, the area of the room. Solve the equation.



Example 4 Continued

Solve3

Let x be the width.Then x + 2 is the length.

Use the formula for area of a rectangle.

l • w = A

length times width = area of room

x + 2 • x = 195



Step 1 x2 + 2x = 195

Step 2

Step 3 x2 + 2x + 1 = 195 + 1

Step 4 (x + 1)2 = 196

Simplify.

Complete the square by

adding 1 to both sides.

Factor the perfect-square

trinomial.

Example 4 Continued

Take the square root of

both sides. Step 5 x + 1 = ± 14

Step 6 x + 1 = 14 or x + 1 = –14 Write and solve two

equations.x = 13 or x = –15

.



Negative numbers are not reasonable for length, so x = 13 is the only solution that makes sense.

The width is 13 feet, and the length is 13 + 2, or 15, feet.

Example 4 Continued

Look Back4

The length of the room is 2 feet greater than the width. Also 13(15) = 195.




An architect designs a rectangular room with an area of 400 ft2. The length is to be 8 ft longer than the width. Find the dimensions of the room. Round your answers to the nearest tenth of a foot.

1 Understand the Problem

The answer will be the length and width of the room.

List the important information:

• The room area is 400 square feet.

• The length is 8 feet more than the width.



2 Make a Plan

Set the formula for the area of a rectangle equal to 400, the area of the room. Solve the equation.

Check It Out! Example 4 Continued



Solve3

Let x be the width.Then x + 8 is the length.

Use the formula for area of a rectangle.

l • w = A


length times width = area of room

X + 8 • x = 400



Step 1 x2 + 8x = 400

Step 3 x2 + 8x + 16 = 400 + 16

Step 4 (x + 4)2 = 416

Simplify.

Complete the square by

adding 16 to both sides.Factor the perfect-

square trinomial.

Step 2


Take the square root of

both sides.

Step 5 x + 4 ± 20.4

Step 6 x + 4 20.4 or x + 4 –20.4

Write and solve two

equations.x 16.4 or x –24.4

.



Negative numbers are not reasonable for length, so x 16.4 is the only solution that makes sense.

The width is approximately16.4 feet, and the length is 16.4 + 8, or approximately 24.4, feet.


Look Back4

The length of the room is 8 feet longer than the width. Also 16.4(24.4) = 400.16, which is approximately 400.




1. x2 +11x +

2. x2 – 18x +


3. x2 – 2x – 1 = 0

4. 3x2 + 6x = 144

5. 4x2 + 44x = 23

Lesson Quiz: Part I

81

6, –8



Lesson Quiz: Part II

6. Dymond is painting a rectangular banner for a football game. She has enough paint to cover 120 ft2. She wants the length of the banner to be 7 ft longer than the width. What dimensions should Dymond use for the banner?

8 feet by 15 feet

Documents

Warm Up Lesson Presentation Lesson Quiz€¦ · Holt Algebra 1 Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 1. Holt McDougal Algebra 1 8-8 Completing the Square Warm