confidential 1 completing the square completing the square

CONFIDENTIAL 1

CompletingCompletingthe Squarethe Square

CONFIDENTIAL 2

Warm UpWarm Up

Solve. Round to the nearest hundredth.

1) 12 = 5x2

2) 3x2 - 4 = 15

3) x2 - 7 = 19

CONFIDENTIAL 3

You have solved quadraticequations by isolating x2 and then using square

roots. This method works if the quadratic equation,when written in standard form, is a perfect square.

Completing the SquareCompleting the Square

When a trinomial is a perfect square, there is arelationship between the coefficient of the x-term

and the constant term.

CONFIDENTIAL 4

x2 + 6x + 9 x2 - 8x +16

62

2 -82

2 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 .

CONFIDENTIAL 5


WORDS NUMBERS ALGEBRA

To complete the square of x2 + bx, add

to the expression. This will form a perfect square

trinomial.

x2 + 6x + __

x2 + 6x +

x2 + 6x + 9

(x + 3)2

x2 + bx + __

x2 + bx +

x2 + bx + 9

(x + b)2

2

62

2 62

2b2

2

CONFIDENTIAL 6


Complete the square to form a perfect square trinomial.

A) x2 + 10x + __

x2 + 10x

B) x2 - 9x + __

x2 + 9x

-92

2102

2= 52 = 25

814

=

x2 + 10x + 25 x2 - 9x + 814

Identify b.

Find b2

2

Add

to the expression

b2

2

CONFIDENTIAL 7

Now you try!


1a. x2 + 12x + __

1b. x2 - 5x + __

1c. 8x + x2 __

CONFIDENTIAL 8

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.

CONFIDENTIAL 9

Solving a Quadratic Equation by Completing the Square

Step1: Write the equation in the form x2 + bx = c.

Step2: Find .

Step3: Complete the square by adding to both sides of the equation.

Step4: Factor the perfect-square trinomial.

Step5: Take the square root of both sides.

Step6: Write two equations, using both the positive and negative square root, and solve each equation.

b2

2

b2

2

CONFIDENTIAL 10

Solving Solving x2 + bx = c by Completing the Square+ bx = c by Completing the Square

Solve by completing the square.

A) x2 + 14x = 15

Step1: x2 + 14x = 15 The equation is in the form x2 + bx = c.

Step2: 142

2 = 72 = 49 Find b2

2

Complete the square.Step3: x2 + 14x + 49 = 15 + 49

Factor and simplify.Step4: (x + 7)2 = 64

Take the square root of both sides.

Step5: x + 7 = ±8

Next page

CONFIDENTIAL 11

Write and solve two equations.

Step6: x + 7 = 8 or x + 7 = -8x = 1 or x = -15

The solutions are 1 and -15.

Check

x2 + 14x = 15

(1)2 + 14(1) 15

1 + 14 15

15 15

x2 + 14x = 15

(-15)2 + 14(-15) 15

225 – 210 15

15 15

CONFIDENTIAL 12

B) x2 - 2x - 2 = 0

Step1: x2 - 2x - 2 = 0 The equation is in the form x2 + bx = c.

Step2: -22

2 = (-1)2 = 1 Find b2

2

Complete the square.Step3: x2 - 2x + 1 = 2 + 1

Factor and simplify.Step4: (x - 1)2 = 3


Step5: x - 1 = ±√3


Step6: x - 1 = √3 or x - 1 = - √3 x = 1 + √3 or x = 1 - √3

The solutions are 1 + √3 and 1 - √3 .

CONFIDENTIAL 13

Now you try!


2a. x2 + 10x = -9

2b. t2 - 8t - 5 = 0

CONFIDENTIAL 14

Solving aSolving ax2 + bx = c by Completing the Square+ bx = c by Completing the Square


A) -2x2 + 12x - 20 = 0

Step1: -2x2 + 12x - 20 = 0 2 2 2

Write in the form x2 + bx = c.

Step2: -62

2 = (-3)2 = 9 Find b2

2

Divide by -2 to make a = 1.

x2 - 6x + 10 = 0

x2 - 6x = -10

Complete the square.Step3: x2 - 6x + 9 = -10 + 9

Factor and simplify.Step4: (x - 3)2 = -1

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

CONFIDENTIAL 15

B) 3x2- 10x = -3

Step1: 3x2 - 10x = -3 3 3 3

Rewrite using likedenominators.

Find b2

2

Divide by 3 to make a = 1.

x2 - 10x = -1 3

x2 + (-10x) + 1 = 0 3

Step2: -10. 1 3 2

2= = 100 = 25 36 9

-10 6

2

Step3:

x2 + (-10x) + 25 = -9 + 25 3 9 9 9

x2 + (-10x) + 25 = -1 + 25 3 9 9

Complete the square.

Next page

CONFIDENTIAL 16


Factor and simplify.

Step5: x – 5 = ± 4 3 3

The solutions are 3 and 1. 3

Step4: x – 5 = 16 3 9

2

Step6: x – 5 = - 4 or x – 5 = - 4 3 3 3 3


x = 3 or x = 1 3

CONFIDENTIAL 17

Now you try!


3a. 3x2 - 5x - 2 = 0

3b. 4t2 - 4t + 9 = 0

CONFIDENTIAL 18

Problem-Solving ApplicationProblem-Solving Application

A landscaper is designing a rectangular brick patio. She has enough bricks to cover 144 square feet. She wants the length

of the patio to be 10 feet greater than the width. What dimensions should she use for the patio? Round to the nearest

hundredth of a foot.

There are enough bricks to cover 144 square feet.

One edge of the patio is to be 10 feet longer than the other edge.

Set the formula for the area of a rectangle equal to 144, the area of the patio. Solve the equation.

Let x be the width.

Then x + 10 is the length.

CONFIDENTIAL 19

Use the formula for area of a rectangle, l × w = A

(x + 10 ) x = 144


Step2: 102

2 = 52 = 25 Find b2

2




Step5: x + 5 = ±13

Next page

CONFIDENTIAL 20


Step6: x + 5 = 13 or x + 5 = -13

x = 8 or x = -18

Negative numbers are not reasonable for length, so x = 8 is the only solution that makes sense.

The width is 8 feet, and the length is 8 + 10, or 18, feet.

The length of the patio is 10 feet greater than the width. Also, 8 (18) = 144.

CONFIDENTIAL 21

Now you try!

Solve using square roots. Check your answer.

4. 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.

CONFIDENTIAL 22

Assessment

1 )x2 + 14x __ +


2) x2 - 4x + __

3) x2 - 3x + __

CONFIDENTIAL 23


6) x2 + x = 30

5) x2 - 8x = 9

4) x2 + 6x = -5

CONFIDENTIAL 24

9) x2 + 16x = 92

7) x2 + 2x = 21

8) x2 - 10x = -9


CONFIDENTIAL 25

10) The length of a rectangle is 4 meters longerthan the width. The area of the rectangle is 80 square

meters. Find the length and width. Round your answers to the nearest tenth of a meter.

CONFIDENTIAL 26


WORDS NUMBERS ALGEBRA

To complete the square of x2 + bx, add

to the expression. This will form a perfect square

trinomial.

x2 + 6x + __

x2 + 6x +

x2 + 6x + 9

(x + 3)2

x2 + bx + __

x2 + bx +

x2 + bx + 9

(x + b)2

2

62

2 62

2b2

2

Let’s review

CONFIDENTIAL 27



A) x2 + 10x + __

x2 + 10x

B) x2 - 9x + __

x2 + 9x

-92

2102

2= 52 = 25

814

=

x2 + 10x + 25 x2 - 9x + 814

Identify b.

Find b2

2

Add

to the expression

b2

2

CONFIDENTIAL 28

Solving a Quadratic Equation by Completing the Square

Step1: Write the equation in the form x2 + bx = c.

Step2: Find .

Step3: Complete the square by adding to both sides of the equation.

Step4: Factor the perfect-square trinomial.

Step5: Take the square root of both sides.

Step6: Write two equations, using both the positive and negative square root, and solve each equation.

b2

2

b2

2

CONFIDENTIAL 29

Solving Solving x2 + bx = c by Completing the Square+ bx = c by Completing the Square


A) x2 + 14x = 15


Step2: 142

2 = 72 = 49 Find b2

2




Step5: x + 7 = ±8

Next page

CONFIDENTIAL 30


Step6: x + 7 = 8 or x + 7 = -8x = 1 or x = -15

The solutions are 1 and -15.

Check

x2 + 14x = 15

(1)2 + 14(1) 15

1 + 14 15

15 15

x2 + 14x = 15

(-15)2 + 14(-15) 15

225 – 210 15

15 15

CONFIDENTIAL 31

B) x2 - 2x - 2 = 0

Step1: x2 - 2x - 2 = 0 The equation is in the form x2 + bx = c.

Step2: -22

2 = (-1)2 = 1 Find b2

2

Complete the square.Step3: x2 - 2x + 1 = 2 + 1

Factor and simplify.Step4: (x - 1)2 = 3


Step5: x - 1 = ±√3


Step6: x - 1 = √3 or x - 1 = - √3 x = 1 + √3 or x = 1 - √3

The solutions are 1 + √3 and 1 - √3 .

CONFIDENTIAL 32

Solving aSolving ax2 + bx = c by Completing the Square+ bx = c by Completing the Square


A) -2x2 + 12x - 20 = 0

Step1: -2x2 + 12x - 20 = 0 2 2 2

Write in the form x2 + bx = c.

Step2: -62

2 = (-3)2 = 9 Find b2

2

Divide by -2 to make a = 1.

x2 - 6x + 10 = 0

x2 - 6x = -10

Complete the square.Step3: x2 - 6x + 9 = -10 + 9

Factor and simplify.Step4: (x - 3)2 = -1

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

CONFIDENTIAL 33

Problem-Solving ApplicationProblem-Solving Application

A landscaper is designing a rectangular brick patio. She has enough bricks to cover 144 square feet. She wants the length

of the patio to be 10 feet greater than the width. What dimensions should she use for the patio? Round to the nearest

hundredth of a foot.

There are enough bricks to cover 144 square feet.

One edge of the patio is to be 10 feet longer than the other edge.

Set the formula for the area of a rectangle equal to 144, the area of the patio. Solve the equation.

Let x be the width.

Then x + 10 is the length.

CONFIDENTIAL 34

Use the formula for area of a rectangle, l × w = A

(x + 10 ) x = 144


Step2: 102

2 = 52 = 25 Find b2

2




Step5: x + 5 = ±13

Next page

CONFIDENTIAL 35


Step6: x + 5 = 13 or x + 5 = -13

x = 8 or x = -18

Negative numbers are not reasonable for length, so x = 8 is the only solution that makes sense.

The width is 8 feet, and the length is 8 + 10, or 18, feet.

The length of the patio is 10 feet greater than the width. Also, 8 (18) = 144.

CONFIDENTIAL 36

You did a great job You did a great job today!today!

confidential 1 completing the square completing the square

Documents