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

[autodidude]

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

 

[haruspex]

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.

 

[Mandlebra]

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

 

[autodidude]

Hmmm...I guess that makes sense...I'll think about it, thanks.

 

[CellCoree]

for sharing your derivation of the formula for lengths of a rectangle. It's always important to understand the reasoning behind mathematical formulas.

The reason why the formula gives both lengths of a rectangle is because it takes into account the fact that a rectangle can have two different sets of dimensions that result in the same perimeter and area. In other words, there are two different solutions to the simultaneous equations A = xy and P = x + y.

Let's break down the formula to see how it works. First, we have the expression [P ± √(P^2 - 16A)]/4. The ± symbol indicates that there are two possible solutions - one with a plus sign and one with a minus sign. The expression inside the square root, (P^2 - 16A), represents the discriminant of the quadratic equation 2y^2 + Py + 2A = 0.

When the discriminant is positive, we get two real solutions for y, which correspond to the two different lengths of the rectangle. When the discriminant is zero, we get one real solution, which means the rectangle is a square. And when the discriminant is negative, we get two imaginary solutions, which do not correspond to the dimensions of a physical rectangle.

So, in short, the formula gives both lengths of a rectangle because it considers all possible solutions to the simultaneous equations and accounts for the different scenarios that can arise. I hope this helps clarify your understanding of the formula.