Explanation:
x = length
y = width
x×y = 48 ft²
y = x - 2
using the second in the first equation we get
x × (x - 2) = 48
x² - 2x - 48 = 0
we can solve this quickly by finding the factors of -48 that are only 2 apart in their absolute value, and the negative factor must be 2 larger in its absolute value than the positive one.
simply because
(a + b)(a - c) = a² - ac + ab - bc
-bc = -48
-ac + ab = -2x
a = x
-c + b = -2
and so we see, 48 = 6×8 (2 difference between the factors).
the larger number must be negative, so, -8×6
and we get
(x + 6)(x - 8)
therefore, the solutions are
x = -6 and x = 8
a negative length does not makes sense here, so x = 8 is or solution.
the length of the garden is 8 ft.
the width is 8 - 2 = 6 ft.