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Finding the Dimensions of a Rectangular Garden with Area 15 Square Feet
Apply the seven-step geometry problem-solving strategy to a rectangle area problem where one dimension is expressed in terms of the other, producing a quadratic equation that must be solved by factoring.
Problem: A rectangular garden has an area of square feet. The length of the garden is two feet more than the width. Find the length and width of the garden.
- Read: A rectangular garden has sq ft, and its length is feet more than its width. Draw and label the rectangle with width and length .
- Identify: The length and width of the garden.
- Name: Let = the width of the garden. Then = the length of the garden.
- Translate: Write the area formula and substitute:
- Solve: Distribute: . Subtract from both sides to obtain standard form: . Factor the trinomial by finding two numbers whose product is and whose sum is : the pair and works. So . Apply the Zero Product Property: or , giving or . Since width cannot be negative, discard . Therefore . The length is .
- Check: Area = sq ft ✓. The length is indeed two more than the width ✓.
- Answer: The width of the garden is feet and the length is feet.
Unlike earlier rectangle problems that used the perimeter formula and produced linear equations, this problem uses the area formula with one dimension expressed relative to the other. Substituting the expression for the length into creates the product , which expands into a quadratic expression . After rearranging to standard form, the equation is solved by factoring and applying the Zero Product Property. The negative solution is algebraically valid but must be discarded because a physical width cannot be negative.
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