The original area of the square is 64 square feet
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Let x be the original side length of the square. The original area is therefore going to be x*x = x^2. For now we don't know the area but once we figure out x, we can compute the area.
The original side length x is increased by 7 ft to jump up to (x+7) ft
At the same time, the other dimension is reduced to (x-2) ft, which is a decrease of 2 ft
The old square is x ft by x ft with area x*x = x^2
The new rectangle is (x+7) ft by (x-2) ft
Let's find the area of this rectangle
Have a look at the image attachment to see how to multiply out (x+7)*(x-2) using the box method
The terms in red are x^2, 7x, -2x and -14 which are found by multiplying the headers
x times x = x^2
x times 7 = 7x
-2 times x = -2x
-2 times 7 = -14
The like terms are 7x and -2x which combine to 5x
Overall the result of multiplying out (x+7)*(x-2) is x^2+5x-14
Note: we can get the same result if we use the FOIL or distribution rule. The box method is a handy way to visually keep track of all the terms.
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The area of the rectangle, in terms of x, is x^2+5x-14
We are told that the area is 90 square feet
So equate the two expressions and solve for x
x^2+5x-14 = 90
x^2+5x-14-90 = 90-90
x^2+5x-104 = 0
(x-8)(x+13) = 0 ... see note below
x-8 = 0 or x+13 = 0
x = 8 or x = -13
note: find two numbers that multiply to -104 and add to 5. Those numbers are -8 and 13 which helps us factor
We ignore x = -13 as a negative side length makes no sense
The only practical root is x = 8
The side length of the original square is 8 feet
The area is
x*x = 8*8 = 64 square feet
We can check the answer by adding 7 to 8 to get 15, and subtracting 2 from 8 to get 6. The 8 by 8 square turns into a 15 by 6 rectangle. Since 15*6 = 90, we have confirmed the right answer.
