Question

Owen’s rectangular garden has a length of (x+3) feet and a width of (x−4) feet. If the area of the garden is 18 square feet, what are the dimensions of the garden?

Answer 1

length: 9 units width: 2 units

Explanation:

The formula to find the area of a rectangle is length*width (LW)

x+3 is the length and x-4 is the width; together they multiply up to 18

(x+3)(x-4) = 18

x^2 - 4x + 3x - 12 = 18 (multiply it out using FOIL)

x^2 - x - 12 = 18 (combine like terms on the left side)

x^2 - x -30 = 0 (subtract 18 from both sides so that the right side is 0)

Think what two numbers multiply up -30 and add up to -1 (5 and -6)

(x+5)(x-6) = 0 (factor it)

x+5 = 0 and x-6 = 0 (solve using zero-product property)

x = -5 and x = 6 (solve)

x = 6 (only this one works since you cannot have a negative side length)

(6+3)(6-4) = 18 ? (plug in to find side lengths and check)

9 * 2 = 18 (Correct!)

Answer 2

length: 9 ft

width: 2 ft

Explanation:

This can be worked using a quadratic equation, or it can be worked by considering factors of the area value.

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difference in dimensions

The difference in length between the long side (x+3) and the short side (x-4) is ...

(x +3) -(x -4) = 3+4 = 7 . . . . feet

area factors

The area formula is ...

A = LW

The given area is 18 square feet, so we want to find numbers that multiply to give 18 and that differ by 7 in their values. Here are the integer factors of 18:

18 = 1×18 = 2×9 = 3×6

Of these factor pairs, the values 2 and 9 differ by 7.

The garden is 9 feet long and 2 feet wide.

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Additional comment

We can check to see if these numbers are consistent with the given expressions:

x+3 = 9 ⇒ x = 6

x -4 = 6 -4 = 2 . . . . the other dimension we found

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If you solve this using a quadratic equation, you will find that dimensions -2 ft and -9 ft also pop out. That is x +3 = -2, or x = -5 is also a solution to the quadratic. Of course, that is an extraneous solution, which we avoid by considering only positive factors of 18.

The quadratic equation we're referring to is A=18 ⇒ (x+3)(x-4) = 18.