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

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Discussion Overview

The discussion revolves around the formula [P±√(P^2-16A)]/4, which is derived from simultaneous equations related to the perimeter and area of a rectangle. Participants explore why this formula yields two different lengths, questioning the implications of the solutions in the context of rectangle dimensions.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant inquires why the formula gives both lengths of a rectangle, suggesting that it should only yield one dimension based on the variable chosen.
  • Another participant notes that the equations do not distinguish between x and y, indicating that if (x, y) is a solution, then (y, x) is also valid.
  • A third participant suggests there may be a typo in the derivation and mentions a potential sign error in the quadratic equation.
  • A later reply expresses understanding and appreciation for the clarification provided.

Areas of Agreement / Disagreement

Participants generally agree on the nature of the solutions being interchangeable, but there is some disagreement regarding the correctness of the derivation and potential errors in the quadratic formula application.

Contextual Notes

Some assumptions about the relationships between perimeter, area, and the dimensions of the rectangle may not be fully explored, and the discussion does not resolve the correctness of the derivation or the presence of any typos.

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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
 
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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.
 
haruspex said:
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.

I believe its a typo, his derivation is correct in the quadratic formula, the quadratic had a sign error also
 
Hmmm...I guess that makes sense...I'll think about it, thanks.
 

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