Remodeling is a Lengthy Process

Bea Hind is remodeling her square family room

by tearing down one wall and extending the room by 3 feet. The room will then be rectangular with

an area of 180 square feet. What are the

current dimensions of the family room?

Bea Hind is remodeling her square family room by tearing down one wall and extending the room by 3 feet. The room will then be rectangular with an area

of 180 square feet. What are the current dimensions of the family room?

Bea’s Original Family Room Ad

ditio

n

3 .ft

3 .ft

= the length of one side of original room

x

x

x

x

x ( ) ( )( 3) ( ) 180

lengthof thenew room widthof thenew room areaof thenew roomx x

2 3 180x x 2 3 180 0x x

( 15)( 12) 0x x

15 0 12 0x x 15 12x x However, we cannot have a

negative length to a room!

12 feet × 12 feet

Remember to check your answer :(12 3)(12) 18015 12 180

Spears and FedX Need More SpaceOnce upon a time when

Brittany and her husband found out they were having another baby, they needed more space

in their square living room. They decided to tear down one wall and extend the room by 2

feet. The room will then be rectangular with an area of 63

square feet. What are the current dimensions of the ex-

Sears’ living room?

Once upon a time when Brittany and her husband found out they were having another baby, they needed more space in their square living room. They decided to tear down

one wall and extend the room by 2 feet. The room will then be rectangular with an area of 63 square feet. What are the current dimensions of the ex-Sears’ living room?

Brittany’s Original Family Room Ad

ditio

n

2 .ft

2 .ft

= the length of one side of original room

x

x

x

x

x ( ) ( )( 2) ( ) 63

lengthof thenew room widthof thenew room areaof thenew roomx x

2 2 63x x 2 2 63 0x x

( 9)( 7) 0x x

9 0 7 0x x 9 7x x However, we cannot have a

negative length to a room!

7 feet × 7 feet

Remember to check your answer :(7 2)(7) 639 7 63

You still gotta work…J Lo is still tutoring math

to earn extra money. Her ex-boyfriend, P. Diddy,

tests her math ability by giving her the following

problem to solve:The square of a number

minus twice the number is 48. Find the number.

The square of a number minus twice the number is 48. Find the number.

n number

A number squared minus twice the number is 48.

2 2 48n n 2 2 48 0n n

( 6)( 8) 0n n

6 0 8 0n n 6 8n n

Check your answer:2( 6) 2( 6) 48 0

36 12 48 0

2(8) 2(8) 48 0 64 16 48 0

You still gotta work…continuedDiddy cannot believe J Lo

correctly answered his problem. He decided to

give her a more challenging problem. He

tells her that:The square of a number is twice the number minus one. Find the number.

The square of a number is twice the number minus one. Find the number.

n number

A number squared is two times the number minus one.

2 2 1n n 2 2 1 0n n

( 1)( 1) 0n n

1 0 1 0n n 1 1n n

Check your answer:2(1) 2(1) 1 0

1 2 1 0

1n

The Return of…You still gotta workDiddy is truly amazed at J Lo’s

mathematical ability as she correctly answered his second problem. He can’t believe that

he ever let such a talented mathematician get away! He decides to give her one last problem. He tells her that:

The sum of the squares of two consecutive odd positive

integers is 650. She must find both integers.

The sum of the squares of two consecutive odd positive integers is 650. She must find both integers.

(odd integer)2 + (next odd integer)2 = 650

First odd integer = Next odd integer = 2 1n 2 3n

2 2(2 1) (2 3) 650n n

(2 1)(2 1) (2 3)(2 3) 650n n n n 2 2(4 2 2 1) (4 6 6 9) 650n n n n n n

2 21( ) ( )4 4 54 1 6 02 9n nn n 2 16 00 658 1n n

5650 6 0 28 16 640 0n n

28( 2 80) 0n n

8( 8)( 10) 0n n

8 0 8 0 10 0n n 8 10n n

2(8)+1 = 17 2(8)+3 = 19

Not a positive integer.

Geometry Genius

Betty Bye is designing a triangular piece of art.

The base of the triangle is 2 inches less

than the height. The area is 40 in2. Find the length of the height and base.

Betty Bye is designing a triangular piece of art. The base of the triangle is 2 inches less than the height. The area is

40 in2. Find the length of the height and base.

h = height

h

2h

2b hArea

( 2)( )40 2h h

2 240 2h h

280 2h h 20 2 80h h

0 ( 10)( 8)h h

10 0 8 0h h 10 8h h

Height = 10 in.

Cannot have a negative length.

2 2

Base = 10 - 2 = 8 in.

Challenge!!!R. U. Nutts has a rectangular

piece of cardboard that is twice as long as it is wide. Squares that are 2 cm on

each side are cut out of each corner, and the ends are turned up to make a box

whose volume is 480 cm3. What are the original

dimensions of the piece of cardboard?

R. U. Nutts has a rectangular piece of cardboard that is twice as long as it is wide. Squares that are 2 cm on

each side are cut out of each corner, and the ends are turned up to make a box whose volume is 480 cm3.

What are the original dimensions of the piece of cardboard?2

2

22

2

22

22x

2x

xx 4x

2 4x

x = width of cardboard

Volume length width height 480 (2 4)( 4)(2)x x

2480 (2 1 (28 ) )4 6xx x 2 12480 (2 16)(2)

2 2xx

2240 2 12 16x x

240 240 20 2 12 224x x 20 2( 6 112)x x

0 4( 14)( 8)x x 0 4 14 0 8 0x x

14 8x x

Width = 14 cmLength = 2(14) = 28 cm