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Zorluk: ZorSolving Quadratic Equations by Factoring

A rectangular garden is surrounded by a uniform gravel path that is 11 foot wide. The length of the garden is 33 feet less than twice its width. If the total area of the garden and the path combined is 117117 square feet, what is the width of the garden, in feet?

Cevap: 7 feet

Cevap

The width of the garden is 7 feet.
By representing the garden's width as ww, the length is 2w−32w - 3. The combined dimensions including the 1-foot uniform path on all sides are w+2w + 2 and 2w−12w - 1. Setting their product equal to the combined area of 117 square feet gives (w+2)(2w−1)=117(w+2)(2w-1) = 117, which simplifies to the quadratic equation 2w2+3w−119=02w^2 + 3w - 119 = 0. Factoring this equation yields (2w+17)(w−7)=0(2w + 17)(w - 7) = 0, giving the solutions w=−8.5w = -8.5 and w=7w = 7. Since width must be positive, the width of the garden is 77 feet.

Adım Adım Çözüm

1
Define variables for the garden's dimensions and the combined dimensions including the path.
Garden width = ww, garden length = 2w−32w - 3. Combined width = w+2w + 2, combined length = 2w−12w - 1.
The path surrounds the garden uniformly, adding 11 foot of width to each of the four sides (adding 22 feet total to both overall width and overall length).
2
Write the area equation for the combined area.
(w+2)(2w−1)=117(w + 2)(2w - 1) = 117
The total area of the garden and path combined is given as 117117 square feet.
3
Expand and rearrange the equation into standard quadratic form aw2+bw+c=0aw^2 + bw + c = 0.
2w2+3w−119=02w^2 + 3w - 119 = 0
Expanding (w+2)(2w−1)(w + 2)(2w - 1) gives 2w2+3w−22w^2 + 3w - 2. Subtracting 117117 from both sides yields the standard form.
4
Factor the quadratic equation over the integers.
(2w+17)(w−7)=0(2w + 17)(w - 7) = 0
We find two numbers that multiply to 2×(−119)=−2382 \times (-119) = -238 and sum to 33. These numbers are 1717 and −14-14. Rewriting the middle term and factoring by grouping yields (2w+17)(w−7)=0(2w + 17)(w - 7) = 0.
5
Solve for ww and select the mathematically and physically valid solution.
w=7w = 7 (discarding the negative root w=−8.5w = -8.5)
A physical measurement like width must be positive.

Anahtar Kavram

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Solving Quadratic Equations by Factoring

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