Part 1
x = side length of the original square
x^2 = area of the original square
One side doubles from x to 2x, while another side goes from x to x-3 (decreases by 3). The rectangle that forms is 2x by x-3 with area 2x(x-3) = 2x^2-6x
This new rectangle has area 25% greater than that of the original square. What this means is that
Area of rectangle = 1.25(area of square)
The 1.25 multiplier means "25% more". You can think of it as 125% or 100% + 25%
So we have...
Area of rectangle = 1.25(area of square)
2x^2-6x = 1.25(area of square)
2x^2-6x = 1.25(x^2)
2x^2-6x = 1.25x^2
as one of the equations that we can use to find the value of x
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Part 2
The equation models the situation because 2x^2-6x on the left side is the area of the rectangle, which came from 2x(x-3)
The right side is 1.25 times x^2, indicating 25% more of the area of the square. You can write it out as 1x^2+0.25x^2 but it leads back to 1.25x^2
Overall, 2x^2-6x = 1.25x^2 tells us "the area of the rectangle is 25% more than the area of the square"
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Part 3
Let's solve for x
2x^2-6x = 1.25x^2
2x^2-6x-1.25x^2 = 0
0.75x^2-6x = 0
x(0.75x-6) = 0
x = 0 or 0.75x-6 = 0
x = 0 or 0.75x = 6
x = 0 or x = 6/0.75
x = 0 or x = 8
Ignore the trivial solution x = 0. The original garden side length will be some positive number.
The only practical solution is x = 8. Therefore the original garden square had side lengths of 8 meters
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Note how 8 doubles to 16, and x-3 = 8-3 = 5.
The old square is 8 by 8 (area 8*8 = 64). The new rectangle is 16 by 5 (area 16*5 = 80). The value 80 is 25% more than 64 because 64*1.25 = 80
Put another way, take 25% of 64 and you should get 16, which adds onto 64 to get 64+16 = 80.
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Answer: Area of the new rectangular garden is 80 square meters
