The area of a rectangular piece of land is 280 square meters. If the length of the land was 5 meters less and the width was 1 meter more, the shape of the land would be a square.

Part A: Write an equation to find the width (x) of the land. Show the steps of your work. (5 points)

Part B: What is the width of the land in meters? Show the steps of your work. (5 points)

original width --- x

original length --- y

xy= 280

y = 280/x

new width --- x+1
new length --- y-5
new area
= (x+1)(y-5)
= (x+1)(280/x - 5)
= 280 - 5x + 280/x - 5
= 275 - 5x + 280/x
= (275x - 5x^2 + 280)/x
= -5( x^2 - 55x - 56)/x

perfect squares in the neighbourhood of 280 are
169 , 196 , 225 , 256 , 289, 324 ...

So I tried values of x starting at x = 13
Lo, and behold, when x = 14
new area = -5(14^2 - 55(14) - 56)/14 = 225 which is a perfect square.

so the original was 14 by 20 for an area of 280

check:
new width = 15
new length = 15
new area = 225
YEAHHH!

Part A:

Let's assume the width of the rectangular piece of land is x meters.
So, the length of the rectangular piece of land would be (x + 5) meters.

According to the problem, the area of the rectangular piece of land is 280 square meters.
The formula to find the area of a rectangle is:
Area = length * width

So, we can write the equation as:
280 = (x + 5) * x

Part B:
To find the width of the land, we need to solve the equation from Part A.

Expanding the equation, we get:
280 = x^2 + 5x

Rearranging the equation, we have:
x^2 + 5x - 280 = 0

This is a quadratic equation, and we can solve it by factoring, completing the square, or using the quadratic formula.
In this case, factoring may not be straightforward, so let's use the quadratic formula:

The quadratic formula is:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)

For our quadratic equation:
a = 1
b = 5
c = -280

Substituting these values into the quadratic formula, we get:
x = (-5 ± sqrt((5^2) - 4(1)(-280))) / (2 * 1)

Simplifying this equation, we have:
x = (-5 ± sqrt(25 + 1120)) / 2
x = (-5 ± sqrt(1145)) / 2

Since the width of the land cannot be negative, we only consider the positive square root:
x = (-5 + sqrt(1145)) / 2

Using a calculator, we find that:
x ≈ 18.9 meters

Therefore, the width of the land is approximately 18.9 meters.