radical Rectangle?

Question

The length of a rectangle is 4 inches more than the width. The number of square inches in its area is twice the number of inches in its perimeter. Find the area of the rectangle. Express your answer in simplified radical form - don't use any rounded decimals. This is what I have so far
8 +4x = Perimeter (4+x) x = Area
2(8 + 4x) = (4+x) x
16 + 8x = 4x + x^2
x^2 -4x-16 =0
x^2 -4x =16
x(x-4) =16
x = 16/x-4

(16/x-4)(16/X-4) +4 =AREA

Answer 1

I would solve the quadratic for x, and then calculate the area.

x^2 -4x-16 =0
x = ( 4 +/- sqrt(16 + 64) ) /2
x = 2 +/- sqrt(4 + 16)
x = 2 +/- 2 sqrt(5)
x = 2 + 2 sqrt(5) (the other root is negative)

I have length: 2 + 2sqrt(5)
Then width = 6 + 2sqrt(5)

Area = length * width
Area = (2 + 2sqrt(5))(6 + 2sqrt(5))
Area = 12 + 12sqrt(5) + 4sqrt(5) + 20
Area = 32 + 16sqrt(5)

Just to check:

Perimeter = 2 * length + 2 * width
Perimeter = 2 * (2 + 2sqrt(5)) + 2 * (6 + 2sqrt(5))
Perimeter = 4 + 4sqrt(5) + 12 + 4sqrt(5)
Perimeter = 16 + 8sqrt(5)

That's, in inches, half of the area in square inches.

Answer 2

I think you lost your way in the third to last line.

width = x
length = x + 4
P = 4x + 8
A = x^2 + 4x = 8x + 16
x^2 - 4x - 16 = 0
x = 2 +/- sqrt(20)
= 2*[1 + sqrt(5)]

A = 8x + 16
= 16 * [2 + sqrt(5)]

NOTE that in this case, it is much easier to compute twice the perimeter, than to multiply length by width. Either way, you'll get hte above a nswer in the most simplified form.

Answer 3

the component to a rectangle is length * width, so, you will do: ?15 * ?20 which equals: ?3 hundred The sq. root of three hundred isn't rational, so which you will no longer provide an precise decimal answer, yet you are able to simplify the unconventional all the way down to: 10?3, or 10 situations the sq. root of three.

Answer 4

l=w+4
A=2p
A=2(2w+2l)
A=2(2w+2w+8)
A=8w+16
A=l*w
A=(w+4)w
A=w^2+4w
w^2+4w=A=8w+16
w^2-4w-16=0
complete the square
w^2-4w=16
w^2-4w+4=16+4
(w-2)^2=20
w-2=sqrt20=2+2sqrt5
A=l*w
A=w^2+4w
A=(2+2sqrt5)^2+4(2+2sqrt5)
A=4+8sqrt5+20+8+8sqrt5
A=32+16sqrt5