Question

the length and width of a rectangle are consecutive even integers. the area of the rectangle is 120 square units. what are the length and width ?

Answer 1

you need to know a few things:
1) what is the formula for the area of a rectangle?

Answer 2

a=wl but i have to solve it with quadratic equations...

Answer 3

2) what does "consecutive even integers" mean ?

Answer 4

they have to be even ?

Answer 5

yes, but more than that. they are "next to" each other. examples:
2 and 4 (we skip the 3 which is odd)
or 6 and 8

and so on

Answer 6

We don't know what the consecutive integers are, so we have to pick a variable name to represent them.

We can do it two ways:
pick x and x+2 to stand for the two consecutive integers.
(of course, this could stand for two consecutive odd numbers... but if we solve the problem we will probably find two even numbers... it's a made up problem and they know we will get even numbers)

or we can force the numbers to be even by using
2x and 2x+2 2x will be even and so will 2x+2

Answer 7

if we use
2x and 2x+2
the problem will be slightly easier to solve. So lets do that.

we don't know which is the length and which the width, but we can just randomly choose:
w= 2x and L= 2x+2

use those definitions in your formula for the area. Can you do that ?
Can you write down the formula?

Answer 8

2x(2x+2)=120 ?

Answer 9

yes. now distribute the 2x. what do you get ?

Answer 10

4x^2+2x=120 ?

Answer 11

now put it in "standard form" (all terms on the left side)

Answer 12

4x^2+4x-120=0

Answer 13

sorry i messed up on the other equation lol

Answer 14

although you should check your "distribute the 2x step"
oh... you have it correct!

Answer 15

now it's always good if you can factor a constant out of each term (if you can) to simplify the quadratic. In this case, divide all terms by 4.
and to keep things equal, divide the other side (the 0) by 4

Answer 16

\[4x ^{2}+4x-120=4*(x-5)*(x+6)\]

Answer 17

and
4(x−5)(x+6) =0

if we simplify (divide both sides by 4)
(x−5)(x+6) =0

and now we have either (x-5)=0 or x+6= 0

solve for x

then use
w= 2x and L= 2x+2

Answer 18

x=5
x=-6 ?

Answer 19

yes, and
w= 2x and L= 2x+2

Answer 20

so i just simplify that ? or leave it as it is ?

Answer 21

we want numbers for the width and length

Answer 22

2,4 ?

Answer 23

we (or rather, you) found
x=5
x=-6

use those values in
w= 2x and L= 2x+2

Answer 24

i dont get it D:

Answer 25

yes, we'll review in a minute. in the meantime, us x=5 in
w= 2x and L= 2x+2

Answer 26

https://in.answers.yahoo.com/question/index?qid=20110603095911AAxjh12

Answer 27

lol 12 and 10

Answer 28

@phi ,@joftheworld see the link

Answer 29

OH ! i get is now ! lol

Answer 30

it*

Answer 31

and just to be complete, what do we get for w and L when x= -6?

Answer 32

just multiply 5 and -6 by 2 ?

Answer 33

but wouldnt it be 10, -12 ?

Answer 34

replace x with -6 in
w= 2x and L= 2x+2

Answer 35

10,10 ?

Answer 36

or 10,-10 ?

Answer 37

slow down.... just erase the x, put in (times) -6 in
w= 2x and L= 2x+2

Answer 38

w=2(5) >>> 10
L=2(-6)+2 >>>-10

Answer 39

oh. no, we don't use different x's to find w and L
we use the *same* x to get w and L

we already used x=5 to find w= 2*5 =10 and L= 2*5+2 = 12

now we are seeing what we get for w and L when we use x=-6

Answer 40

-12, -10 ?

Answer 41

yes, and notice with w= -12 and L=-10
w*L= -12* -10= +120
so this "answer" works... but we don't use negative numbers for lengths... we ignore this "answer" (sometimes called an "extraneous solution" )

the other pair, w=10 and L=12 is the answer.
(though we don't know if it's w=12 and L=10 ... though I think of the width as being shorter than the length in a rectangle)

Answer 42

thanks can you help with couple more ? it should be easy..

Answer 43

the length of a rectangle plus its width is 18 cm, the area is 80 square cm. what is the length and width ?

Answer 44

im guessing its 10,8 ? because 80/8=10 ? and 10+8 is 18 ..

Answer 45

Now to review:
area of the rectangle is 120

we used that to write
w*L= 120

length and width of a rectangle are consecutive even integers

this part is a bit tricky. To get an *even* number, we write 2x
2x will always be even, no matter what x is. (this is a "trick" people learn)

if we have 2x, the next integer up is 2x+2
(btw, we could have gone down, and picked 2x-2... that will work)

I said let w be the first number, w= 2x
(going on the theory that the width should be smaller than the length.. but it does not have to be)
and let
L be the other number, L= 2x+2

we then said w*L= 120
2x(2x+2) =120
from which we get
4x^2 +4x -120=0
or
x^2 +x -30=0
(x+6)(x-5) =0

x= 5 or x=-6
we use x=5 in w=2x and L=2x+2 (notice we use the same x, or we would not get consecutive even numbers for w and L)
we get
w=10 and L=12

Answer 46

did you read what i said above ?

Answer 47

I just did.

yes you have the answer. Now can you solve it using equations?

Answer 48

yes but i have to hurry up and do this lesson .. i have to be somewhere soon

Answer 49

length of a rectangle plus its width is 18 cm

write down the formula this is saying

Answer 50

10*8=80

Answer 51

whoops sorry 10+8=18

Answer 52

yes, if you know the answer, you can do that.

but if you don't know the numbers, how do you write it?

Answer 53

x(x+1)=80 ?

Answer 54

are you good with functional values ?

Answer 55

if you don't have numbers, you use names (letters) instead
in fact, it's the same thing as
length of a rectangle plus its width is 18 cm
but "shortened" by replacing the long words with short names.