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Formula for lengths of a rectangle - why does it give both lengths?

  1. Jun 19, 2012 #1
    Why does the formula [P±√(P^2-16A)]/4 give the values of either of two different lengths of a rectangle? (P is perimeter and A is area)

    I derived it by solving two simultaneous equations, A = xy and P=x+y and then applying the quadratic formula to the resulting second-order equation 2y^2 + Py+2A thus getting y=[P±√(P^2-16A)]/4

    I tried out some numbers just to test it out and was surprised that both solutions were lengths of the rectangle, so it gave y but also x…I fail to see how this so, shouldn't it only give the length y (or x, if I'd eliminated y instead when I solved the simultaneous equations)?

    Thanks
     
  2. jcsd
  3. Jun 20, 2012 #2

    haruspex

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    The equations cannot 'know' which is x and which is y. If x = A, y = B is a solution then so is x = B, y = A.
    Btw, the perimeter would be 2x+2y.
     
  4. Jun 20, 2012 #3
    I believe its a typo, his derivation is correct in the quadratic formula, the quadratic had a sign error also
     
  5. Jun 20, 2012 #4
    Hmmm...I guess that makes sense...I'll think about it, thanks.
     
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