Question

A rectangular vegetable garden will have a width that is 2 feet less than the length, and an area of 48 square feet. If x represents the length, then the length can be found by solving the equation:

What is the length, x, of the garden?

Answers

  1. Step-by-step 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.

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