Question 165121
Draw a diagram of the problem -- it'll help you see how to solve it.
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If you do, the "inner rectangle" represents the garden while the "outer rectangle" (bordered by the outside edge of the gravel path).
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"outer rectangle" - "inner rectangle" = "area of gravel path"
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Let x = width of gravel path
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"outer rectangle" then is:
(2x+18)(2x+13) 
= 4x^2 + 26x + 36x + 234
= 4x^2 + 62x + 234
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"Inner rectangle" is:
18 * 13 = 234
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Now, instead of:
"outer rectangle" - "inner rectangle" = "area of gravel path"
we have:
(4x^2 + 62x + 234) - 234 = 516
4x^2 + 62x = 516
4x^2 + 62x - 516 = 0
2x^2 + 31x - 258 = 0
Since, it is difficult to factor, use the quadratic equation.  It will yield:
x = {6, -21.5}
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Since the negative solution does not make sense, throw it out leaving:
x = 6 feet (width of gravel path)
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Details of quadratic:
*[invoke quadratic "x", 2, 31, -258 ]