Question

The distance from one corner of a rectangular garden to the other is 13 ft. The length of the garden is 7 ft longer than the width. Write a quadratic equation to find the dimension of the garden. Solve the equation and find the area of the garden in square feet.

Answer 1

• dimensions: 12 ft by 5 ft
• area: 60 ft²

Explanation:

Let x represent the shorter dimension in feet. Then the longer one is x+7 and the Pythagorean theorem tells us the relation of these to the diagonal is ...

x² + (x+7)² = 13²

2x² +14x + 49 = 169 . . . . eliminate parentheses

x² +7x -60 = 0 . . . . . subtract 169 and divide by 2

(x +12)(x -5) = 0 . . . . factor the equation

x = -12 or +5 . . . . . . . only the positive value of x is useful here.

The short dimension is 5 ft, so the long dimension is 12 ft. The area is their product, 60 ft².

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Comment on finding the area

The quadratic equation above can be rearranged and factored as ...

x(x +7) = 60

Since the dimensions of the garden are x and (x+7), this product is the garden's area. This equation tells us the area is 60. We don't actually have to find the dimensions.