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I tried this with an area of $157m^2$ but got the wrong answer of $0.7334m$ when the correct answer was $1.138m$. Now I'm trying it with $140m^2$ I wanted to know what might be the issue with my steps below?

A garden plot must have a central planting area of length $13m$ and width $8m$. There is to be a sidewalk around its edge of width $w$. If the total area, planting area plus sidewalk area, is $140m^2$, what is the sidewalk width $w$ in meters?

$$(13 + 2w)(8 + 2w) = 140$$ $$4w^2 + 42w + 104 = 140$$ $$4w^2 + 42w - 36 = 0$$ We need to use the quadratic fomula to find the answer. $${x = \frac{{ - b \pm \sqrt {b^2 - 4ac} }}{{2a}}}$$ $$(2w + 0.79669)(2w - 11.29669) = 0$$ $$2w = 11.29669$$ $$w = \frac{11.29669}{2}$$ $w$ can only be positive so we end up with $5.648345$

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Everything up to the Quadratic Formula is fine. Why did you divide by $2$? It was $w$, not a mysterious $x$, that the formula finds. I suggest doing that calculation again, it will give you $w$. Be careful about signs. –  André Nicolas Aug 30 '12 at 3:20

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You are not using the quadratic formula correctly. The quadratic formula does not tell you that you can factor it as $$(2w + 0.79669)(2w - 11.29669) = 0$$

What it tells you are the roots are 0.79669 and -11.29669. As you said, the answer must be positive, so $w = 0.79669.$

If you want to factor it, the correct factoring is $$4(w-0.79669)(w+11.29669) = 0$$

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