Using Factoring to Sove Problems Help?

Question

A garden plot 4 meters by 12 meters has one side along a fence on the top. The area of the garden is to be doubled by digging a border of uniform width on the other three sides. What should the width of the border be? A picture can be seen at
http://mathassessments.mscenters.org/view_test.php?Mode=Add&GradeLevelID=9
Number 6 under the opened end section

Answer 1

The area currently is 48 meters ( 4 * 12), so in order to double it the area on the outside also have to add up to 48 meters.

There are three rectangles that make up the area on the sides, and the area of a rectangle is width * length. So if we add up the area of the three rectangles it has to add up to 48.

All three rectangle have the same width = x, but the length are different according to which side it's on. There are two 4 meters sides, and one 12 meters sides.

So the equation is...

4 * x + 4 * x + 12 * x = 48
20 * x = 48
x = 48/20 = 12/5 meters

Answer 2

Ok let's see what we know. We know we have an initial area of 48 square meters (4*12). If we double that, we get 96 square meters. We have to add another 48 square meters. But that area comes from three separate imaginary rectangles, and they aren't of equal area. However, two of the rectangles are equal in area because the width and length are the same. The length for each of these (on the side) is:
4+x, where x is the width, so we have an Length of the two equal to: (4+x)
Their area is the width(x) times Length:
x(4+x)=4x+x^2
Then we have a longer rectangle, which is equal to 12, and its area is 12x.
Since we have an area of 48 square meters, we have:
4x+x^2+4x+x^2+12x=48
20x+2x^2=48, or 2x^2+20x-48=0
we factor out the 2:
x^2+10x-24=0
To solve for x we can try factoring out terms, or we can use the quadratic formula. Intuitively we can guess at the answer here, which is 2. But factoring gives us:
(x-2)(x+12)=0
We can never have an area of -12, so clearly +2 (for x) is our border width. Check this by plugging in 2.

Answer 3

So, to start, the garden has an area of 48 square units; however, after doubling the area it will have an area of 96 square units. The new width will be 12+2x and the new height will be 4 +x. Now, here is where the factoring comes in, since the area of a rectangle is still base times height, the area of 96=(12+2x)(4+x). Next multiply this out to get 96=48+20x+2x^2; divide both sides by 2 to get 48=24+10x+x^2; next subtract 48 from both sides to get x^2+10x-24=0; finally, factor this to get (x+12)(x-2)=0. There are two answers for this: x=-24 and x=2--obviously x=2 is the answer because it makes no sense to expand the boundary of the garden by -12. Hope that helped.